I would like to know how small private colleges in New York state
define the steps of "progressive discipline" of its employees, and if
any college there must have these steps written down in a formal
document in order for this policy to be implemented fairly and legally.

Dear Stan Laurel, 

I went through "small private colleges" in New York and found no
published document on the subject. I did find many documents relating
to SUNY and NYU/ Cornell policies, if you'd like.

Sort of by definition as "small and private" you are unlikely to find
anything without contacting these institutions directly, or by
perusing their web sites. Even then there is no guarantee that they
will have promulgated the policies you seek. Public institutions
usually have state and federal requirements regarding disclosure of
policies, while private institutions generally have less restrictions.

I doubt if this information is compiled online in one place.  I
suggest you contact the New York State United Teachers (a union that
is attempting to organize faculty at some private colleges) to see if
they have information.
http://www.nysut.org/