I think thats right, using the slang baby as a direct object is rare but possible. “I shot my baby” e.g., but using it with “another baby” really throws it off because a lover is not something you usually have multiple of at the same time so there is not enough context to resolve the ambiguity and lots of natives default to the wrong meaning.

But the comma would make _baby_ refer to the listener as an addressee of the singer (vocative). That is not the meaning the article says is appropriate. Commas are not put between a determiner and its noun so a comma would give a third meaning, turning _another_ into a pronoun.