Both sentences you provide make sense to me, a native AmE speaker. And I think either still make sense if you swap for a vacation with for vacation However you might choose one of the other for a certain emphasis:
Using a vacation creates an image of a general single discrete leisure trip. This should make sense with the typical usage of the a article.
Using vacation can cause it to act as a descriptor, instead of a noun. In this case it could be replaced with a word like leisure Like in your sentence, depending on the context it could sound like:
I'd like to travel to New York for vacation (as opposed to for work).
Using vacation should also be used when talking in more general indefinite terms that could span multiple trips. Using this might conjure up more aristocratic feelings, like using the verb form of summer.
I like going to New York for
In terms of the question you linked, here is my interpretation:
I personally don't think any of the preposition choices are always better, it is very context dependent. For example, using parallel structures by repeating for a creates a perfectly good sentence:
I went to Italy last year for a business trip, and went back a year later for a vacation.
You can also replace for a with on :
I went to Italy last year on business
trip, and went back a year later on vacation.
It all depends on which way you are trying to emphasize. If you just want to emphasize that the Italy visit was leisurely, say "for vacation"/"on vacation". For this sentence if you include "a" this time it makes it sound like:
"I went to Italy for (one of my) vacation(s).
Overall .. Either sound okay to my native ears and have similar meanings. But you might want to use one over the other depending on the connotation you would want to convey.