According to 14 CFR Part 97.3 (b), it provides the speed ranges for
the different aircraft categories (A-E). In all the documentation I've
read, this "speed" is the IAS of the airplane.

And so it is (almost - I seem to recall it's really CAS, but that
wouldn't make much difference). That's the regulation.

However, my CFI says that this is based on the Ground Speed.

It's hard to prove a negative, so I can't say there is NO regulatory
support for what he says, but I've certainly never seen it. Have you
asked him to show you where he read this? Further, without RNAV that
works at low altitudes or DME on the approach (which isn't rare but is
far from universal), ground speed is an estimate - and these rules are
a lot older than widespread use of RNAV that works at low altitudes.

In other words - I think your CFI is totally wrong on this one.

The reason I ask is that I've been asked questions before where the
examiner was trying to make sure that I completely understood the
rule, and I'm worried that selecting minimums that are higher than
necessary will show that I don't really understand it.

Well, yes, it will. Only I think you understand it fine; it's your
instructor who is steering you wrong.

There are situations where it makes sense to select higher minimums on
an approach (especially a circling approach) where the higher speed
makes remaining within the protected area for the lower mins
problematic. I certainly don't think it would be wrong to say "Yes, I
know that technically cat A mins apply, but I am going to use Cat B
mins because the wind conditions make remaining within the Cat A
protected area problematic." If the situation is a circling approach
with restrictions imposed and very high winds that would require an
excessive bank angle to remain within the protected area, he would
probably consider that a sign of good judgment. But you should be
clear that this is something you are choosing to do because it makes
sense, and that the regulations do permit lower mins.

Michael