Sorry for answering my own question, but I'd like to add a complement to Joe Silveman's answer (for those who may be interested), which however is too long to fit into a comment.

1. A reference to the "two-sided natural density" on $\mathbf Z$ is implicit to [2, Chapter 3, Section 5], where the notion is used to count the number of elements, integers, or units of a number field with height bounded above by some $b \in \mathbf R^+$.

2. I strongly agree with those arguing that a symmetric definition of the upper asymptotic density on $\mathbf Z$ is much more natural than the asymmetric definition, but I can't really mention a single paper or book on additive theory where the former is used (which may be partially due to the fact that "most" authors in this field work with densities defined on the power set of $\mathbf N^+$).

3. There is a relatively nice way to recover both definitions as a special case of a more general construction. To see how, let $\mathbf H$ denote either $\mathbf Z$, $\mathbf N$, or $\mathbf N^+$, fix $\alpha \in [-1,\infty[$, and take $\mathfrak{F} = (F_n)_{n \ge 1}$ to be a sequence of nonempty finite subsets of $\mathbf H$ such that $F_n \ne \{0\}$ for every $n$ and there exists a doubly indexed sequence $(\theta_{h,k})_{h, k \ge 1}$ of nondecreasing functions $\mathbf N^+ \to \mathbf N^+$ with the property that, for all $h, k \in \mathbf N^+$, the image of $\theta_{h,k}$ contains every sufficiently large positive integer and the following hold: $$ \lim_{n \to \infty} \frac{\sum_{i \in \Delta_n(\mathfrak{F},\theta_{h,k})} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = 0 \qquad\text{and}\qquad \lim_{n \to \infty} \frac{\sum_{i \in F_{\theta_{h,k}(n)}} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = \frac{1}{k^{\alpha+1}}, $$ where $\Delta_n(\mathfrak{F},\theta_{h,k})$ denotes, for each $n$, the symmetric difference of the sets $\mathbf H \cap (k^{-1} \cdot (F_n - h))$ and $ F_{\theta_{h,k}(n)}$. Next, consider the function $$\mathsf d^\ast(\mathfrak{F};\alpha): \mathbf H \to \mathbf R: X \mapsto \limsup_{n \to \infty} \frac{\sum_{i \in X \cap F_n} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha}. $$ It is not difficult to verify that all the above conditions are satisfied, e.g., if $F_n := [\![1, n^q ]\!]$ for some $q \in \mathbf N^+$, or $\alpha \ge 0$ and $F_n := [\![ na, nb ]\!]$ for some $a,b \in \mathbf H$ with $a < b$.

This is but a definition, and not even a very appealing one. But things get perhaps more interesting if we consider the following proposition (those interested may give a look at [3, Proposition 4] for a proof):

Proposition. $\mathsf d^\ast(\mathfrak{F};\alpha)$ is an upper density.

Here an upper density (on $\mathbf H$) is a function $\mathcal P(\mathbf H) \to \mathbf R$ such that, for all $X,Y \subseteq \mathbf H$ and $h,k \in \mathbf N^+$, the following hold:

1. $\mu^\ast(\mathbf H) = 1$;
2. $\mu^\ast(X) \le \mu^\ast(Y)$ whenever $X \subseteq Y$;
3. $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$;
4. $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$.

So, based on the proposition above, and in continuity with [1, Definition 1.4], which covers the special case where $\mathbf H = \mathbf N^+$ and $F_n = [\![ 1, n ]\!]$, we may call $\mathsf d^\ast(\mathfrak{F};\alpha)$ the upper $\alpha$-density (on $\mathbf H$) relative to $\mathfrak{F}$.

In fact, $\mathsf d^\ast(\mathfrak{F};0)$ boils down to the "two-sided" (respectively, "one-sided") asymptotic density referred to in the OP in the special case where $\mathbf H = \mathbf Z$ and $F_n := [\![-n,n]\!]$ (respectively, $F_n := [1, n]$) for all $n$.

Bibliography.

[1] R. Giuliano-Antonini and G. Grekos, Comparison between lower and upper α-densities and lower and upper $\alpha$-analytic densities, Unif. Distrib. Theory 3 (2008), No. 2, 21-35.

[2] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.

[3] P. Leonetti and S.T., On the notions of upper and lower density, preprint (arXiv:1506.04664)

Sorry for answering my own question, but I'd like to add a complement to Joe Silveman's answer (for those who may be interested), which however is too long to fit into a comment.

1. A reference to the "two-sided natural density" on $\mathbf Z$ is implicit to [2, Chapter 3, Section 5], where the notion is used to count the number of elements, integers, or units of a number field with height bounded above by some $b \in \mathbf R^+$.

2. I strongly agree with those arguing that a symmetric definition of the upper asymptotic density on $\mathbf Z$ is much more natural than the asymmetric definition, but I can't really mention a single paper or book on additive theory where the former is used (which may be partially due to the fact that "most" authors in this field work with densities defined on the power set of $\mathbf N^+$).

3. There is a relatively nice way to recover both definitions as a special case of a more general construction. To see how, let $\mathbf H$ denote either $\mathbf Z$, $\mathbf N$, or $\mathbf N^+$, fix $\alpha \in [-1,\infty[$, and take $\mathfrak{F} = (F_n)_{n \ge 1}$ to be a sequence of nonempty finite subsets of $\mathbf H$ such that $F_n \ne \{0\}$ for every $n$ and there exists a doubly indexed sequence $(\theta_{h,k})_{h, k \ge 1}$ of nondecreasing functions $\mathbf N^+ \to \mathbf N^+$ with the property that, for all $h, k \in \mathbf N^+$, the image of $\theta_{h,k}$ contains every sufficiently large positive integer and the following hold: $$ \lim_{n \to \infty} \frac{\sum_{i \in \Delta_n(\mathfrak{F},\theta_{h,k})} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = 0 \qquad\text{and}\qquad \lim_{n \to \infty} \frac{\sum_{i \in F_{\theta_{h,k}(n)}} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = \frac{1}{k^{\alpha+1}}, $$ where $\Delta_n(\mathfrak{F},\theta_{h,k})$ denotes, for each $n$, the symmetric difference of the sets $\mathbf H \cap (k^{-1} \cdot (F_n - h))$ and $ F_{\theta_{h,k}(n)}$. Next, consider the function $$ \mathsf d^\ast(\mathfrak{F};\alpha): \mathcal P(\mathbf H) \to \mathbf R: X \mapsto \limsup_{n \to \infty} \frac{\sum_{i \in X \cap F_n} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha}. $$ It is not difficult to verify that all the above conditions are satisfied, e.g., if $F_n := [\![1, n^q ]\!]$ for some $q \in \mathbf N^+$, or $\alpha \ge 0$ and $F_n := [\![ na, nb ]\!]$ for some $a,b \in \mathbf H$ with $a < b$.

This is but a definition, and not even a very appealing one. But things get perhaps more interesting if we consider the following proposition (those interested may give a look at [3, Proposition 4] for a proof):

Proposition. $\mathsf d^\ast(\mathfrak{F};\alpha)$ is an upper density.

Here an upper density (on $\mathbf H$) is a function $\mathcal P(\mathbf H) \to \mathbf R$ such that, for all $X,Y \subseteq \mathbf H$ and $h,k \in \mathbf N^+$, the following hold:

1. $\mu^\ast(\mathbf H) = 1$;
2. $\mu^\ast(X) \le \mu^\ast(Y)$ whenever $X \subseteq Y$;
3. $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$;
4. $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$.

So, based on the proposition above, and in continuity with [1, Definition 1.4], which covers the special case where $\mathbf H = \mathbf N^+$ and $F_n = [\![ 1, n ]\!]$, we may call $\mathsf d^\ast(\mathfrak{F};\alpha)$ the upper $\alpha$-density (on $\mathbf H$) relative to $\mathfrak{F}$.

In fact, $\mathsf d^\ast(\mathfrak{F};0)$ boils down to the "two-sided" (respectively, "one-sided") asymptotic density referred to in the OP in the special case where $\mathbf H = \mathbf Z$ and $F_n := [\![-n,n]\!]$ (respectively, $F_n := [\![1, n]\!]$) for all $n$.

Bibliography.

[1] R. Giuliano-Antonini and G. Grekos, Comparison between lower and upper α-densities and lower and upper $\alpha$-analytic densities, Unif. Distrib. Theory 3 (2008), No. 2, 21-35.

[2] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.

[3] P. Leonetti and S.T., On the notions of upper and lower density, preprint (arXiv:1506.04664)

Sorry for answering my own question, but I'd like to add a complement to Joe Silveman's answer (for those who may be interested), which however is too long to fit into a comment.

1. A reference to the "two-sided natural density" on $\mathbf Z$ is implicit to [2, Chapter 3, Section 5], where the notion is used to count the number of elements, integers, or units of a number field with height bounded above by some $b \in \mathbf R^+$.

2. I strongly agree with those arguing that a symmetric definition of the upper asymptotic density on $\mathbf Z$ is much more natural than the asymmetric definition, but I can't really mention a single paper or book on additive theory where the former is used (which is partially due to the fact that "most" authors in this field work with densities defined on the power set of $\mathbf N^+$).

3. There is a relatively nice way to recover both definitions as a special case of a more general construction. To see how, let $\mathbf H$ denote either $\mathbf Z$, $\mathbf N$, or $\mathbf N^+$, fix $\alpha \in [-1,\infty[$, and take $\mathfrak{F} = (F_n)_{n \ge 1}$ to be a sequence of nonempty finite subsets of $\mathbf H$ such that $F_n \ne \{0\}$ for every $n$ and there exists a doubly indexed sequence $(\theta_{h,k})_{h, k \ge 1}$ of nondecreasing functions $\mathbf N^+ \to \mathbf N^+$ with the property that, for all $h, k \in \mathbf N^+$, the image of $\theta_{h,k}$ contains every sufficiently large positive integer and the following hold: $$ \lim_{n \to \infty} \frac{\sum_{i \in \Delta_n(\mathfrak{F},\theta_{h,k})} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = 0 \qquad\text{and}\qquad \lim_{n \to \infty} \frac{\sum_{i \in F_{\theta_{h,k}(n)}} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = \frac{1}{k^{\alpha+1}}, $$ where $\Delta_n(\mathfrak{F},\theta_{h,k})$ denotes, for each $n$, the symmetric difference of the sets $\mathbf H \cap (k^{-1} \cdot (F_n - h))$ and $ F_{\theta_{h,k}(n)}$. Next, consider the function $$\mathsf d^\ast(\mathfrak{F};\alpha): \mathbf H \to \mathbf R: X \mapsto \limsup_{n \to \infty} \frac{\sum_{i \in X \cap F_n} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha}. $$ It is not difficult to verify that all the above conditions are satisfied, e.g., if $F_n := [\![1, n^q ]\!]$ for some $q \in \mathbf N^+$, or $\alpha \ge 0$ and $F_n := [\![ na, nb ]\!]$ for some $a,b \in \mathbf H$ with $a < b$.

This is but a definition, and not even a very appealing one. But things get perhaps more interesting if we consider the following proposition (those interested may give a look at [3, Proposition 4] for a proof):

Proposition. $\mathsf d^\ast(\mathfrak{F};\alpha)$ is an upper density.

Here an upper density (on $\mathbf H$) is a function $\mathcal P(\mathbf H) \to \mathbf R$ such that, for all $X,Y \subseteq \mathbf H$ and $h,k \in \mathbf N^+$, the following hold:

1. $\mu^\ast(\mathbf H) = 1$;
2. $\mu^\ast(X) \le \mu^\ast(Y)$ whenever $X \subseteq Y$;
3. $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$;
4. $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$.

So, based on the proposition above, and in continuity with [1, Definition 1.4], which covers the special case where $\mathbf H = \mathbf N^+$ and $F_n = [\![ 1, n ]\!]$, we may call $\mathsf d^\ast(\mathfrak{F};\alpha)$ the upper $\alpha$-density (on $\mathbf H$) relative to $\mathfrak{F}$.

In fact, $\mathsf d^\ast(\mathfrak{F};0)$ boils down to the "two-sided" (respectively, "one-sided") asymptotic density referred to in the OP in the special case where $\mathbf H = \mathbf Z$ and $F_n := [\![-n,n]\!]$ (respectively, $F_n := [1, n]$) for all $n$.

Bibliography.

[1] R. Giuliano-Antonini and G. Grekos, Comparison between lower and upper α-densities and lower and upper $\alpha$-analytic densities, Unif. Distrib. Theory 3 (2008), No. 2, 21-35.

[2] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.

[3] P. Leonetti and S.T., On the notions of upper and lower density, preprint (arXiv:1506.04664)

Sorry for answering my own question, but I'd like to add a complement to Joe Silveman's answer (for those who may be interested), which however is too long to fit into a comment.

1. A reference to the "two-sided natural density" on $\mathbf Z$ is implicit to [2, Chapter 3, Section 5], where the notion is used to count the number of elements, integers, or units of a number field with height bounded above by some $b \in \mathbf R^+$.

2. I strongly agree with those arguing that a symmetric definition of the upper asymptotic density on $\mathbf Z$ is much more natural than the asymmetric definition, but I can't really mention a single paper or book on additive theory where the former is used (which may be partially due to the fact that "most" authors in this field work with densities defined on the power set of $\mathbf N^+$).

3. There is a relatively nice way to recover both definitions as a special case of a more general construction. To see how, let $\mathbf H$ denote either $\mathbf Z$, $\mathbf N$, or $\mathbf N^+$, fix $\alpha \in [-1,\infty[$, and take $\mathfrak{F} = (F_n)_{n \ge 1}$ to be a sequence of nonempty finite subsets of $\mathbf H$ such that $F_n \ne \{0\}$ for every $n$ and there exists a doubly indexed sequence $(\theta_{h,k})_{h, k \ge 1}$ of nondecreasing functions $\mathbf N^+ \to \mathbf N^+$ with the property that, for all $h, k \in \mathbf N^+$, the image of $\theta_{h,k}$ contains every sufficiently large positive integer and the following hold: $$ \lim_{n \to \infty} \frac{\sum_{i \in \Delta_n(\mathfrak{F},\theta_{h,k})} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = 0 \qquad\text{and}\qquad \lim_{n \to \infty} \frac{\sum_{i \in F_{\theta_{h,k}(n)}} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = \frac{1}{k^{\alpha+1}}, $$ where $\Delta_n(\mathfrak{F},\theta_{h,k})$ denotes, for each $n$, the symmetric difference of the sets $\mathbf H \cap (k^{-1} \cdot (F_n - h))$ and $ F_{\theta_{h,k}(n)}$. Next, consider the function $$\mathsf d^\ast(\mathfrak{F};\alpha): \mathbf H \to \mathbf R: X \mapsto \limsup_{n \to \infty} \frac{\sum_{i \in X \cap F_n} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha}. $$ It is not difficult to verify that all the above conditions are satisfied, e.g., if $F_n := [\![1, n^q ]\!]$ for some $q \in \mathbf N^+$, or $\alpha \ge 0$ and $F_n := [\![ na, nb ]\!]$ for some $a,b \in \mathbf H$ with $a < b$.

This is but a definition, and not even a very appealing one. But things get perhaps more interesting if we consider the following proposition (those interested may give a look at [3, Proposition 4] for a proof):

Proposition. $\mathsf d^\ast(\mathfrak{F};\alpha)$ is an upper density.

Here an upper density (on $\mathbf H$) is a function $\mathcal P(\mathbf H) \to \mathbf R$ such that, for all $X,Y \subseteq \mathbf H$ and $h,k \in \mathbf N^+$, the following hold:

1. $\mu^\ast(\mathbf H) = 1$;
2. $\mu^\ast(X) \le \mu^\ast(Y)$ whenever $X \subseteq Y$;
3. $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$;
4. $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$.

So, based on the proposition above, and in continuity with [1, Definition 1.4], which covers the special case where $\mathbf H = \mathbf N^+$ and $F_n = [\![ 1, n ]\!]$, we may call $\mathsf d^\ast(\mathfrak{F};\alpha)$ the upper $\alpha$-density (on $\mathbf H$) relative to $\mathfrak{F}$.

In fact, $\mathsf d^\ast(\mathfrak{F};0)$ boils down to the "two-sided" (respectively, "one-sided") asymptotic density referred to in the OP in the special case where $\mathbf H = \mathbf Z$ and $F_n := [\![-n,n]\!]$ (respectively, $F_n := [1, n]$) for all $n$.

Bibliography.

[1] R. Giuliano-Antonini and G. Grekos, Comparison between lower and upper α-densities and lower and upper $\alpha$-analytic densities, Unif. Distrib. Theory 3 (2008), No. 2, 21-35.

[2] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.

[3] P. Leonetti and S.T., On the notions of upper and lower density, preprint (arXiv:1506.04664)