Is this statement grammatically correct as-is, or is an extra comma required?

Settle an argument for me. This is the statement as it is written:

An introverted teenage girl with an unconventional superpower, Taylor goes out in costume to find escape from a deeply unhappy and frustrated civilian life.

My friend thinks this is wrong, and there should be a comma after "Taylor", like so:

An introverted teenage girl with an unconventional superpower, Taylor, goes out in costume...

My position is that that would also be correct, but in a different way and it's fine as-is. Am I wrong here?

Both of them are correct punctuation. You win the bet. As a native speaker, you will immediately have a correct human intuition about what the differences are.

The second one emphasizes what her name is.

I will just add a nuance that seems to be present here. Without comma after Taylor, there is more emphasis on "Taylor" as the main subject of the sentence. In this case, some words were omitted:

[Being/As she is] an introverted teenage girl with an unconventional superpower, Taylor goes out in costume to find escape from a deeply unhappy and frustrated civilian life.

An introverted teenage girl with an unconventional superpower, would be a supplement without which the rest of the sentence would still stand:

Taylor goes out in costume to find escape from a deeply unhappy and frustrated civilian life.

Thus, the first part of the sentence until the comma becomes an explanation why Taylor went and did that.

If you add that second comma, Taylor becomes an apposition, a supplementary information. The main subject of the sentence will be the NP An introverted teenage girl with an unconventional superpower, (which happens to be named Taylor). The relation of cause and effect between An introverted teenage girl with an unconventional superpower, Taylor, and goes out in costume to find escape from a deeply unhappy and frustrated civilian life is not marked and so it is lost. In this case it is Taylor that can be omitted without changing the meaning of the sentence:

An introverted teenage girl with an unconventional superpower goes out in costume to find escape from a deeply unhappy and frustrated civilian life.

Therefore, depending on what one means, both versions could be correct. But the fact that the author of this sentence chose not to put a second comma, indicates that he believes Taylor acted in this way because she was an introverted, but at the same time very powerful girl.