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Short answer: The two conditions on $e$ are logically unrelated. For example, a long root vector in type $B_2$ (generating the minimal nonzero nilpotent orbit) satisfies 1) but not 2), while an element in the subregular nilpotent orbit in type $G_2$ satisfies 2) but not 1).

Details. A basic source is the paper [EK] by Elashvili and Kac in a 2006 AMS book series, which Ben reminds me was posted on arXiv here. They classify case-by-case the possible good gradings for $e$. One of these is the Dynkin grading, which is even iff the labels on the Dynkin diagram of $e$ (or its orbit) are all even. In some cases but not others there are other good gradings.

A minimal $e$ in type $B_2$ (indeed all four nilpotent orbits here) is regular/standard in a Levi subalgebra of a parabolic. But its Dynkin diagram has an odd label, so the Dynkin grading is not even. In the fourth paragraph of their Section 2, [EK] point out that except for type $A_n$ the only possible good grading for a minimal $e$ is the Dynkin grading.

On the other hand, Theorem 7.1 in [EK] says that in types $G_2$ and $F_4$ only the Dynkin grading is good for each $e$. A subregular nilpotent in type $G_2$ fails to be of standard Levi type: there are only four types of parabolic subalgebras and four corresponding standard Levi type orbits here, the regular, zero, minimal, and "middle" orbits. But a subregular $e$ has even Dynkin labels, so there is an even good grading for $e$.

Discussion. Sorting out nilpotent orbits with various properties gets quite complicated. For instance, all even orbits are Richardson, but some Richardson orbits are odd, such as the orbit labelled $C_3$ in type $F_4$ (if my old computations of Richardson orbits for $F_4$ are correct). Coupled with Theorem 7.1 in [EK] this gives an odd Richardson element having no even good grading. (But in general, when some even grading does exist with $e \in \mathfrak{g}_2$, Proposition 2.1 in [EK] shows that the grading is good for e if and only if $e$ is Richardson.)

Philosophically, the problem here is that standard Levi type and Richardson are in a sense dual ideas (which coincide only in type $A_n$). While "most" nilpotent orbits are of both types, there are exceptions that make the independent technical conditions 1) and 2) in the question both useful to impose for some purposes in the study of finite $W$-algebras.

Short answer: The two conditions on $e$ are logically unrelated. For example, a long root vector in type $B_2$ (generating the minimal nonzero nilpotent orbit) satisfies 1) but not 2), while an element in the subregular nilpotent orbit in type $G_2$ satisfies 2) but not 1).

Details. A basic source is the paper [EK] by Elashvili and Kac in a 2006 AMS book series, which Ben reminds me was posted on arXiv here. They classify case-by-case the possible good gradings for $e$. One of these is the Dynkin grading, which is even iff the labels on the Dynkin diagram of $e$ (or its orbit) are all even. In some cases but not others there are other good gradings.

A minimal $e$ in type $B_2$ (indeed all four nilpotent orbits here) is regular/standard in a Levi subalgebra of a parabolic. But its Dynkin diagram has an odd label, so the Dynkin grading is not even. In the fourth paragraph of their Section 2, [EK] point out that except for type $A_n$ the only possible good grading for a minimal $e$ is the Dynkin grading.

On the other hand, Theorem 7.1 in [EK] says that in types $G_2$ and $F_4$ only the Dynkin grading is good for each $e$. A subregular nilpotent in type $G_2$ fails to be of standard Levi type: there are only four types of parabolic subalgebras and four corresponding standard Levi type orbits here, the regular, zero, minimal, and "middle" orbits. But a subregular $e$ has even Dynkin labels, so there is an even good grading for $e$.

Discussion. Sorting out nilpotent orbits with various properties gets quite complicated. For instance, all even orbits are Richardson, but some Richardson orbits are odd, such as the orbit labelled $C_3$ in type $F_4$ (if my old computations of Richardson orbits for $F_4$ are correct). Coupled with Theorem 7.1 in [EK] this gives an odd Richardson element having no even good grading. (But in general, when some even grading does exist with $e \in \mathfrak{g}_2$, Proposition 2.1 in [EK] shows that the grading is good for e if and only if $e$ is Richardson.)

Philosophically, the problem here is that standard Levi type and Richardson are in a sense dual ideas (which coincide only in type $A_n$). While "most" nilpotent orbits are of both types, there are exceptions that make the independent technical conditions 1) and 2) in the question both useful to impose for some purposes in the study of finite $W$-algebras.

[EDITED] If I understand your set-up correctly, condition 2) fails to imply condition 1). In the other direction I'm not yet sure about counterexamples. But Sasha Premet can give a more definitive answer, having been a pioneer in the study of finite $W$-algebras. (As Yves comments, it would help to provide more precise background.) I assume you are looking at finite $W$-algebras in the spirit of Losev's 2010 ICM survey here and related literature. Here it's conventional to leave aside the extreme case $e=0$. Many nilpotents are of "standard Levi type": principal (also called regular) in a Levi subalgebra of some parabolic subalgebra of the given simple Lie algebra, but outside type $A_n$ there are other nilpotents as well.

The most standard examples of even gradings of $\mathfrak{g}$ arise directly from the adjoint action of any three-dimensional simple subalgebra containing $e$ and correspond to Dynkin diagrams for nilpotent orbits having only even labels. For type $A_n$ there are sometimes odd labels, but there exist good even gradings for arbitrary $e$ in this unusually well-behaved type.

On the other hand, in other simple types there are examples of even nilpotents which are not of standard Levi type, such as the subregular nilpotents in types $G_2$ and $D_4$. That $G_2$ case has a good even grading (the standard Dynkin grading), so 2) holds here but not 1).

The more general "good" gradings were classified by Elashvili-Kac and developed further by Brundan-Goodwin here. But the connection with parabolics seems to be narrower here than in the notion of standard Levi type. (By the way, it's tricky to sort out in the Dynkin-Kostant classification those nilpotent orbits which are of standard Levi type, as seen already in the subregular case.)

P.S. Although you are interested in characteristic 0, essentially everything remains the same in good prime characteristic.

[ADDED] I still have to double-check the results of Elashvili-Kac at UMass, but I think the simplest example where 1) fails to imply 2) involves the minimal nilpotent orbit in type $B_2$. Here you can take $e$ to be a long simple root vector, which is of standard Levi type. The Dynkin diagram has an odd label, so the Dynkin grading itself isn't even. On the other hand, this should be one of the cases where Elashvili-Kac find no other good grading.

Short answer: The two conditions on $e$ are logically unrelated. For example, a long root vector in type $B_2$ (generating the minimal nonzero nilpotent orbit) satisfies 1) but not 2), while an element in the subregular nilpotent orbit in type $G_2$ satisfies 2) but not 1).

Details. A basic source is the paper [EK] by Elashvili and Kac in a 2006 AMS book series, which Ben reminds me was posted on arXiv here. They classify case-by-case the possible good gradings for $e$. One of these is the Dynkin grading, which is even iff the labels on the Dynkin diagram of $e$ (or its orbit) are all even. In some cases but not others there are other good gradings.

A minimal $e$ in type $B_2$ (indeed all four nilpotent orbits here) is regular/standard in a Levi subalgebra of a parabolic. But its Dynkin diagram has an odd label, so the Dynkin grading is not even. In the fourth paragraph of their Section 2, [EK] point out that except for type $A_n$ the only possible good grading for a minimal $e$ is the Dynkin grading.

On the other hand, Theorem 7.1 in [EK] says that in types $G_2$ and $F_4$ only the Dynkin grading is good for each $e$. A subregular nilpotent in type $G_2$ fails to be of standard Levi type: there are only four types of parabolic subalgebras and four corresponding standard Levi type orbits here, the regular, zero, minimal, and "middle" orbits. But a subregular $e$ has even Dynkin labels, so there is an even good grading for $e$.

Discussion. Sorting out nilpotent orbits with various properties gets quite complicated. For instance, all even orbits are Richardson, but some Richardson orbits are odd, such as the orbit labelled $C_3$ in type $F_4$ (if my old computations of Richardson orbits for $F_4$ are correct). Coupled with Theorem 7.1 in [EK] this gives an odd Richardson element having no even good grading. (But in general, when some even grading does exist with $e \in \mathfrak{g}_2$, Proposition 2.1 in [EK] shows that the grading is good for e if and only if $e$ is Richardson.)

Philosophically, the problem here is that standard Levi type and Richardson are in a sense dual ideas (which coincide only in type $A_n$). While "most" nilpotent orbits are of both types, there are exceptions that make the independent technical conditions 1) and 2) in the question both useful to impose for some purposes in the study of finite $W$-algebras.

[EDITED] If I understand your set-up correctly, condition 2) fails to imply condition 1). In the other direction I'm not yet sure about counterexamples. But Sasha Premet can give a more definitive answer, having been a pioneer in the study of finite $W$-algebras. (As Yves comments, it would help to provide more precise background.) I assume you are looking at finite $W$-algebras in the spirit of Losev's 2010 ICM survey here and related literature. Here it's conventional to leave aside the extreme case $e=0$. Many nilpotents are of "standard Levi type": principal (also called regular) in a Levi subalgebra of some parabolic subalgebra of the given simple Lie algebra, but outside type $A_n$ there are other nilpotents as well.

The most standard examples of even gradings of $\mathfrak{g}$ arise directly from the adjoint action of any three-dimensional simple subalgebra containing $e$ and correspond to Dynkin diagrams for nilpotent orbits having only even labels. For type $A_n$ there are sometimes odd labels, but there exist good even gradings for arbitrary $e$ in this unusually well-behaved type.

On the other hand, in other simple types there are examples of even nilpotents which are not of standard Levi type, such as the subregular nilpotents in types $G_2$ and $D_4$. That $G_2$ case has a good even grading (the standard Dynkin grading), so 2) holds here but not 1).

The more general "good" gradings were classified by Elashvili-Kac and developed further by Brundan-Goodwin here. But the connection with parabolics seems to be narrower here than in the notion of standard Levi type. (By the way, it's tricky to sort out in the Dynkin-Kostant classification those nilpotent orbits which are of standard Levi type, as seen already in the subregular case.)

P.S. Although you are interested in characteristic 0, essentially everything remains the same in good prime characteristic.

[ADDED] I still have to double-check the results of Elashvili-Kac at UMass, but I think the simplest example where 1) fails to imply 2) involves the minimal nilpotent orbit in type $B_2$. Here you can take $e$ to be a lonf simple root vector, which is of standard Levi type. The Dynkin diagram has an odd label, so the Dynkin grading itself isn't even. On the other hand, this should be one of the cases where Elashvili-Kac find no other good grading.

[EDITED] If I understand your set-up correctly, condition 2) fails to imply condition 1). In the other direction I'm not yet sure about counterexamples. But Sasha Premet can give a more definitive answer, having been a pioneer in the study of finite $W$-algebras. (As Yves comments, it would help to provide more precise background.) I assume you are looking at finite $W$-algebras in the spirit of Losev's 2010 ICM survey here and related literature. Here it's conventional to leave aside the extreme case $e=0$. Many nilpotents are of "standard Levi type": principal (also called regular) in a Levi subalgebra of some parabolic subalgebra of the given simple Lie algebra, but outside type $A_n$ there are other nilpotents as well.

The most standard examples of even gradings of $\mathfrak{g}$ arise directly from the adjoint action of any three-dimensional simple subalgebra containing $e$ and correspond to Dynkin diagrams for nilpotent orbits having only even labels. For type $A_n$ there are sometimes odd labels, but there exist good even gradings for arbitrary $e$ in this unusually well-behaved type.

On the other hand, in other simple types there are examples of even nilpotents which are not of standard Levi type, such as the subregular nilpotents in types $G_2$ and $D_4$. That $G_2$ case has a good even grading (the standard Dynkin grading), so 2) holds here but not 1).

The more general "good" gradings were classified by Elashvili-Kac and developed further by Brundan-Goodwin here. But the connection with parabolics seems to be narrower here than in the notion of standard Levi type. (By the way, it's tricky to sort out in the Dynkin-Kostant classification those nilpotent orbits which are of standard Levi type, as seen already in the subregular case.)

P.S. Although you are interested in characteristic 0, essentially everything remains the same in good prime characteristic.

[ADDED] I still have to double-check the results of Elashvili-Kac at UMass, but I think the simplest example where 1) fails to imply 2) involves the minimal nilpotent orbit in type $B_2$. Here you can take $e$ to be a long simple root vector, which is of standard Levi type. The Dynkin diagram has an odd label, so the Dynkin grading itself isn't even. On the other hand, this should be one of the cases where Elashvili-Kac find no other good grading.