I've read Ed's query and Max's reply.  Here are some musings on the issue:
 
I wonder what would have been the case if Ed had asked his tutors/students
about "infinite" verbs or "infinitives."  One would think that surely they have
heard of them; surely they'd have a useful concept linked to one of those terms
So, if "infinite" has meaning, why does a positive version--"finite"--seem so
devoid of meaning?
 
I began a brief search of my limited library of grammar and handbooks.  I was
interested to discover that few spend much time on what "finite" means or what
uses the IDEA of "finite" verbs can be put to.  Max indicates that he does make
good use of it.   I'm very curious to hear what others have done.  I.E.  Have
you made the use Max does?  Have you done so but not as explicitly?
 
I also found it interesting that some handbooks make a distinction between
"infinite" and "nonfinite" verbs.  Yet the distinction seems slippery (if there
at all).
 
"Nonfinite" verbs describe "unfinished actions"; they are "verbals."
"Infinite" doesn't merit its own definition.  But as I mentioned above, and as
one would expect, "Infinitive" does.  An "infinitive" is "a nonfinite verb form
(or verbal)."  The notion that it is a "nonfinite verb form" seems to imply
there are others, thus making "infinitive" a subspecies of "nonfinite."  Or
does it?  What do you think?
 
I'm now very curious whether we need to work to clarify this issue.
 
                        Jim Dubinsky ("a question is a question, no matter
                                       how small"