As a NYC-based dancer, I have two immediate responses:
1. I went to an event last night that featured a free lesson at the beginning from a teacher who has some decent name recognition. He taught what he called NY-style timing with the first step on beat 1. It wasn't just beat 1 in name, either. It was literally beat 1 in the music. That said, dancers here with what I consider real flavor fudge the timing on beat 1 a bit earlier to the 8 or the 'and' of 8. Some people get this, some people don't. Both may claim to dance NY style.
(Side rant: very few teachers (in my experience) call that first step anything other than "1" which is frustrating. They should call it what it is: the 8 or the 'and' of 8. This would make transitioning from On 1 a hell of a lot easier...)
2. Sensitive dancers will adjust the non-stressed beats of their timing (everything except 2 and 6) to fit with the music, both in a general sense, and also for certain measures where one musical point or another catches their interest. (For instance, to me, it doesn't make sense to use the same basic/timing I use for Tito Puente's "Ninas y Senoras" as I would for El Canario's "Pensando toda el dia en ti". Some people might view this as different basics/timings. I view it as dancing to the music, and not to a pattern of steps. (Again, for both songs, I'm still breaking on 2 and 6. But I'll probably step a lot more on beat 1 for the Puente song.)
(Side rant #2: There are plenty of dancers who don't do this and frankly, it can be frustrating to dance with someone who dances the same way to every song, considering how many subgenres you would encounter on a typical dance event, and how many different rhythmic patterns are used within each subgenre.)