If both I and the CFI are club members, who is providing the aircraft - me or
the CFI? (Presumably, I'm paying for it that day.)

Absent contortions to the contrary, you are. You choose the aircraft
and schedule it, you pay for it. You pick the instructor and pay him.
The instructor's being a club =member= is not the same as supplying the
aircraft.

So long as you are free to use another instructor with this aircraft,
and are free to use a non-club aircraft with this instructor, they are
not tied together.

Generally a club does not employ instructors. They permit instructors
to instruct in their aircraft.

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.