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The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and i think this is the case in Mac Lane's "Homology" book as well).
As far as i can understand, the "present day definition" (i.e. the "unsigned" and "ungraded" as presented in most hopf algebra textbooks after the '70's - that is after Sweedler's textbook) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.

It is not my purpose -in this answer- to argue on what is the "correct" definition but rather to try to provide some terminology, unifying the above descriptions: That is the language of braided groups, in the sense this notion has been introduced and used since the mid'90's after the Majid's school of hopf algebras and quantum groups.
Imo, one of the advantages of this language is that it helps in clarifying that the "grading" and the "signs" are not the same thing (the same grading may refer to diferent sign rules).

The ususal Poincare-Birkhoff-Witt theorem generalizes as well.

According to present day terminology (here i am mainly following the terminology as used by the Majid's school) such structures are called braided groups or hopf algebras in the braided monoidal Category ${}_{\mathbb{CZ}_2}\mathcal{M}$ of $\mathbb{CZ}_2$-modules (remember that the $\mathbb{CZ}_2$-modules are exactly the $\mathbb{Z}_2$-graded v.s. or "super"-v.s.). The braiding $\theta$ can be shown to be "generated" through a "1-1" correspondence, with the non-trivial, quasitriangular structure (i.e. the non-trivial $R$-matrix) of the $C\mathbb{Z_2}$ group hopf algebra: $$R=\frac{1}{2}\big(1\otimes 1+1\otimes g+g\otimes 1-g\otimes g\big)$$ (More generally, under the term braided groups we frequently mean hopf algebras in the braided monoidal Category ${}_{H}\mathcal{M}$ of $H$-modules, where $H$ is any quasitriangular hopf algebra -and not necessarilly a group algebra).

It has been shown that given a $(\mathbb{CZ}_{2},\theta)$-hopf algebra $H$ (or: a "super"-hopf algebra $H$) we can form the smash product hopf algebra $H\star \mathbb{CZ}_2$ by adjoining an extra generator $g$ to $H$. This is an ordinary hopf algebra and the construction is functorial in the sense that there is an equivalenve of categories $${}_{H}\underline{\mathcal{M}} \thicksim {}_{H \star CZ_{2}}\mathcal{M}$$ between the Category ${}_{H}\underline{\mathcal{M}}$ of super-reps of $H$ and the Category ${}_{H \star CZ_{2}}\mathcal{M}$ of representations of the (ordinary) smash product Hopf algebra $H \star CZ_{2}$.
The converse procedure can also be done: these are the Bosonization and Transmutation techniques developed in the early 90's (and partially based on the previously known idea of Radford's biproduct).

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and I think this is the case in Mac Lane's "Homology" book as well).
As far as I can understand, the "present day definition" (i.e. the "unsigned" and "ungraded" as presented in most hopf algebra textbooks after the '70's - that is after Sweedler's textbook) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.

It is not my purpose -in this answer- to argue on what is the "correct" definition but rather to try to provide some terminology, unifying the above descriptions: That is the language of braided groups, in the sense this notion has been introduced and used since the mid'90's after the Majid's school of hopf algebras and quantum groups.
Imo, one of the advantages of this language is that it helps in clarifying that the "grading" and the "signs" are not the same thing (the same grading may refer to different sign rules).

The usual Poincare-Birkhoff-Witt theorem generalizes as well.

According to present day terminology (here i am mainly following the terminology as used by the Majid's school) such structures are called braided groups or hopf algebras in the braided monoidal Category ${}_{\mathbb{CZ}_2}\mathcal{M}$ of $\mathbb{CZ}_2$-modules (remember that the $\mathbb{CZ}_2$-modules are exactly the $\mathbb{Z}_2$-graded v.s. or "super"-v.s.). The braiding $\theta$ can be shown to be "generated" through a "1-1" correspondence, with the non-trivial, quasitriangular structure (i.e. the non-trivial $R$-matrix) of the $C\mathbb{Z_2}$ group hopf algebra: $$R=\frac{1}{2}\big(1\otimes 1+1\otimes g+g\otimes 1-g\otimes g\big)$$ (More generally, under the term braided groups we frequently mean hopf algebras in the braided monoidal Category ${}_{H}\mathcal{M}$ of $H$-modules, where $H$ is any quasitriangular hopf algebra -and not necessarily a group algebra).

It has been shown that given a $(\mathbb{CZ}_{2},\theta)$-hopf algebra $H$ (or: a "super"-hopf algebra $H$) we can form the smash product hopf algebra $H\star \mathbb{CZ}_2$ by adjoining an extra generator $g$ to $H$. This is an ordinary hopf algebra and the construction is functorial in the sense that there is an equivalence of categories $${}_{H}\underline{\mathcal{M}} \thicksim {}_{H \star CZ_{2}}\mathcal{M}$$ between the Category ${}_{H}\underline{\mathcal{M}}$ of super-reps of $H$ and the Category ${}_{H \star CZ_{2}}\mathcal{M}$ of representations of the (ordinary) smash product Hopf algebra $H \star CZ_{2}$.
The converse procedure can also be done: these are the Bosonization and Transmutation techniques developed in the early 90's (and partially based on the previously known idea of Radford's biproduct).

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and i think this is the case in Mac Lane's "Homology" book as well).
As far as i can understand, the "present day definition" (i.e. the "unsigned" as presented in most hopf algebra textbooks after the '90's) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.

The UEA $\mathbb{U}(L)$ of the $(G, \theta)$-Lie algebra $L$, is not a hopf algebra -well at least not in the "ordinary" (ungraded) sense (here "ordinary" should be taken to mean the modern day definition of hopf algebras as this is presented in most of the hopf algebra textbooks which have appeared after the '90's). Let me try to shed some light in this point:
$\mathbb{U}(L)$ is equipped with a "comultiplication"
$$ \underline{\Delta} : \mathbb{U}(L) \rightarrow \mathbb{U}(L) \underline{\otimes} \mathbb{U}(L) $$ which is a homomorphism of assoc., $G$-graded algebras (equivalently: a homogeneous homomorphism of assoc algebras of degree $0$) i.e. $$ \underline{\Delta}(ab) = \sum \theta\big(deg(a_{2}), deg(b_{1})\big) a_{1}b_{1} \otimes a_{2}b_{2} = \underline{\Delta}(a) \underline{\Delta}(b) $$ for all $a,b \in \mathbb{U}(L)$, with $\underline{\Delta}(a) = \sum a_{1} \otimes a_{2}$, $\underline{\Delta}(b) = \sum b_{1} \otimes b_{2}$, and $a_{2}$, $b_{1}$ homogeneous. The product $\underline{\Delta}(a) \underline{\Delta}(b)$ in the rhs of the above is understood to be in the $\mathbb{U}(L) \underline{\otimes} \mathbb{U}(L)$ algebra.
Due to the universal property of the UEA, $\underline{\Delta}$ is uniquely defined by its values on the elements of $L$ i.e. on the generators of $\mathbb{U}(L)$ $$ \underline{\Delta}(x) = 1 \otimes x + x \otimes 1 $$ Similarly, $\mathbb{U}(L)$ is equipped with the "antipode" $\underline{S} : U(L) \rightarrow U(L)$ which is no more an algebra antihomomorphism (as in the "ordinary" hopf algebra case) but a twisted or braided antihomomorphism of $G$-graded algebras, in the sense that: $$ \underline{S}(ab) = \theta\big(deg(a), deg(b)\big) \underline{S}(b)\underline{S}(a) $$ and $deg(a) = deg(\underline{S}(a))$ for all homog elements $a,b \in \mathbb{U}(L)$. Again the universal property of the UEA, ensures us that the antipode is uniquely defined by its values on the elements of $L$ i.e. on the generators of the UEA $\mathbb{U}(L)$: $$ \underline{S}(x) = -x $$ If the above are complemented with the counit $\underline{\varepsilon}(x) = 0 $ for all $x \in \mathbb{U}(L)$ then we get a $G$-graded, $\theta$-braided Hopf algebra or -for short- a $(G,\theta)$-hopf algebra.

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and i think this is the case in Mac Lane's "Homology" book as well).
As far as i can understand, the "present day definition" (i.e. the "unsigned" and "ungraded" as presented in most hopf algebra textbooks after the '70's - that is after Sweedler's textbook) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.

The UEA $\mathbb{U}(L)$ of the $(G, \theta)$-Lie algebra $L$, is not a hopf algebra -well at least not in the "ordinary" (ungraded) sense (here "ordinary" should be taken to mean the modern day definition of hopf algebras as this is presented in most of the hopf algebra textbooks which have appeared after the '70's). Let me try to shed some light in this point:
$\mathbb{U}(L)$ is equipped with a "comultiplication"
$$ \underline{\Delta} : \mathbb{U}(L) \rightarrow \mathbb{U}(L) \underline{\otimes} \mathbb{U}(L) $$ which is a homomorphism of assoc., $G$-graded algebras (equivalently: a homogeneous homomorphism of assoc algebras of degree $0$) i.e. $$ \underline{\Delta}(ab) = \sum \theta\big(deg(a_{2}), deg(b_{1})\big) a_{1}b_{1} \otimes a_{2}b_{2} = \underline{\Delta}(a) \underline{\Delta}(b) $$ for all $a,b \in \mathbb{U}(L)$, with $\underline{\Delta}(a) = \sum a_{1} \otimes a_{2}$, $\underline{\Delta}(b) = \sum b_{1} \otimes b_{2}$, and $a_{2}$, $b_{1}$ homogeneous. The product $\underline{\Delta}(a) \underline{\Delta}(b)$ in the rhs of the above is understood to be in the $\mathbb{U}(L) \underline{\otimes} \mathbb{U}(L)$ algebra.
Due to the universal property of the UEA, $\underline{\Delta}$ is uniquely defined by its values on the elements of $L$ i.e. on the generators of $\mathbb{U}(L)$ $$ \underline{\Delta}(x) = 1 \otimes x + x \otimes 1 $$ Similarly, $\mathbb{U}(L)$ is equipped with the "antipode" $\underline{S} : U(L) \rightarrow U(L)$ which is no more an algebra antihomomorphism (as in the "ordinary" hopf algebra case) but a twisted or braided antihomomorphism of $G$-graded algebras, in the sense that: $$ \underline{S}(ab) = \theta\big(deg(a), deg(b)\big) \underline{S}(b)\underline{S}(a) $$ and $deg(a) = deg(\underline{S}(a))$ for all homog elements $a,b \in \mathbb{U}(L)$. Again the universal property of the UEA, ensures us that the antipode is uniquely defined by its values on the elements of $L$ i.e. on the generators of the UEA $\mathbb{U}(L)$: $$ \underline{S}(x) = -x $$ If the above are complemented with the counit $\underline{\varepsilon}(x) = 0 $ for all $x \in \mathbb{U}(L)$ then we get a $G$-graded, $\theta$-braided Hopf algebra or -for short- a $(G,\theta)$-hopf algebra.

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and i think this is the case in Mac Lane's "Homology" book as well).
As far as i can understand, the "present day definition" (i.e. the "unsigned" as presented in most hopf algebra textbooks after the '90's) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.

The notion of hopf algebras has slowly emerged from the work of topologists in the late '30's and '40's on the cohomology of compact Lie groups and their homogeneous spaces. Initially the term had been used in the "graded" or "signed" sense (for example, Milnor and Moore's seminal paper refers to $\mathbb{Z}$-graded hopf algebras and i think this is the case in Mac Lane's "Homology" book as well).
As far as i can understand, the "present day definition" (i.e. the "unsigned" as presented in most hopf algebra textbooks after the '90's) has been formulated during the early '60's and is mainly due to works of Cartier and Dieudonne. (the introduction of A primer of Hopf algebras includes interesting details on that point) However, lots of people from different schools have contributed to the evolution of this notion; lots of authors have kept using the term for signed or graded objects long after that. So it does not come as a surprise that there are still discussions on the "correct" form of the definition.