Choosing full time vs part time on a PhD application, is one necessarily better than the other for increasing chances of admission?
Or does part time indicate that I would have other commitments and so would not be seen as a whole hearted student (and hence full time means that I would be a whole hearted student)?


Yes, this is my understanding of how it would be interpreted.  It would definitely not mean working as a TA or RA.

Unless you have a job you'd like to keep, or other comparable responsibilities outside of studying, I'd recommend checking only the full-time box.  I don't think expressing an openness to part-time study would help get admitted to a math PhD program, while it might look odd or worrisome.

In particular, in the math PhD programs I've studied or taught in, there have been essentially no part-time students.  (It might have been a theoretical option, but I can't think of any actual part-time students.)  For programs like this, the application might ask about it, but the expectation would be that everyone is a full-time student.

I think this is standard in U.S. math PhD programs, but some programs may run differently, so I can't say this advice is necessarily universal.  (The biggest variance would be that some applied math programs may be more open to having part-time students.)