In the literature there appear to be two different definitions of "regular functions":
defined locally by polynomials https://en.wikipedia.org/wiki/Morphism_of_algebraic_varieties
defined locally by rational functions as a well defined quotient of polynomials about each point. http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/chapter-2.pdf
how does changing the definition of regular function affect the theory? Can I just assume the first definition?
If you're trying to define what it means for a function to be regular on an open subset of an affine variety, you must you definition 2: a function is regular on this open subset iff it can locally be written as a ratio of polynomials with non-vanishing denominators.
In the special case where you're defining what it means for a function to be regular on the entire affine variety, this is equivalent to definition 1: a function is regular on the entire affine variety iff it can globally be written as a polynomial. And yes, I do mean "globally", not "locally".