$\{1\}$ is an Aut-basis of the group of profinite integers $\hat{\mathbb Z}$, but since this group has cardinality $2^\omega$, it has no logical generator. However, every element of $\hat{\mathbb Z}$ is type-definable from $1$ (i.e., any pair of distict elements are distinguishable by a first-order formula with parameter $1$); using Ali Enayat’s terminology, the structure $\langle\hat{\mathbb Z},+,1\rangle$ is Leibnizian. (Conversely, if $G$ is the group reduct of a $\mathbb Z$-group such that $\{1\}$ is an Aut-basis of $G$, then all elements of $G$ are type-definable from $1$, and $G$ embeds in $\hat{\mathbb Z}$; see my paper Rigid models of Presburger arithmetic.)

Note that “every element of $G$ is type-definable from $s$” gives you a general property to consider that is intermediate between “$\{s\}$ is an Aut-basis of $G$”, and “$s$ is a logical generator of $G$”.

$\{1\}$ is an Aut-basis of the group of profinite integers $\hat{\mathbb Z}$, but since this group has cardinality $2^\omega$, it has no logical generator. However, every element of $\hat{\mathbb Z}$ is type-definable from $1$ (i.e., any pair of distict elements are distinguishable by a first-order formula with parameter $1$); using Ali Enayat’s terminology, the structure $\langle\hat{\mathbb Z},+,1\rangle$ is Leibnizian. (Conversely, if $G$ is the group reduct of a $\mathbb Z$-group such that $\{1\}$ is an Aut-basis of $G$, then all elements of $G$ are type-definable from $1$, and $G$ embeds in $\hat{\mathbb Z}$; see my paper Rigid models of Presburger arithmetic.)

Similarly, if $p$ is any prime, the group of $p$-adic integers $\mathbb Z_p$ has no logical generator (being uncountable), but all its elements are type-definable from $1$, thus $\{1\}$ is an Aut-basis.

Note that “every element of $G$ is type-definable from $s$” gives you a general property to consider that is intermediate between “$\{s\}$ is an Aut-basis of $G$”, and “$s$ is a logical generator of $G$”.

$\{1\}$ is an Aut-basis of $\hat{\mathbb Z}$, but since this group has cardinality $2^\omega$, it has no logical generator. However, every element of $\hat{\mathbb Z}$ is type-definable from $1$; using Ali Enayat’s terminology, the structure $\langle\hat{\mathbb Z},+,1\rangle$ is Leibnizian. (Conversely, if $G$ is the group reduct of a $\mathbb Z$-group such that $\{1\}$ is an Aut-basis of $G$, then all elements of $G$ are type-definable, and $G$ embeds in $\hat{\mathbb Z}$; see my paper Rigid models of Presburger arithmetic.)

Note that “every element of $G$ is type-definable from $s$” gives you a general property to consider that is intermediate between “$\{s\}$ is an Aut-basis of $G$”, and “$s$ is a logical generator of $G$”.

$\{1\}$ is an Aut-basis of the group of profinite integers $\hat{\mathbb Z}$, but since this group has cardinality $2^\omega$, it has no logical generator. However, every element of $\hat{\mathbb Z}$ is type-definable from $1$ (i.e., any pair of distict elements are distinguishable by a first-order formula with parameter $1$); using Ali Enayat’s terminology, the structure $\langle\hat{\mathbb Z},+,1\rangle$ is Leibnizian. (Conversely, if $G$ is the group reduct of a $\mathbb Z$-group such that $\{1\}$ is an Aut-basis of $G$, then all elements of $G$ are type-definable from $1$, and $G$ embeds in $\hat{\mathbb Z}$; see my paper Rigid models of Presburger arithmetic.)

Note that “every element of $G$ is type-definable from $s$” gives you a general property to consider that is intermediate between “$\{s\}$ is an Aut-basis of $G$”, and “$s$ is a logical generator of $G$”.