I ordered a 2.25" trackball from Ultimarc .... So do I need a 2.25" hole saw?
2.25", yes.
Hole saw, fostner, blade type all work, some better in different materials. Since you're doing metal, I'd suggest a metal cutting hole saw.
Or is that the diameter of the ball itself,
Yes the ball is also 2.25".
making the hole size smaller?
umm, see your next question
I also know there is a small lip around the trackball that looks to be about a quarter of an inch wide. Does that change the hole saw size?
yes. The hole would be smaller, but the lip changes the hole size back to 2.25". And yes, the lip totals ~1/4" (1/8" on each side).
Sorry its a stupid question,
Not a stupid question. You knew enough to understand where all the factors came in, but not how they finally ended up. Which is the perfect reason to ask, after searching of course.
but I couldn't find any answers searching the board (you'd be surprised how often hole saw and trackball are posted).
Not really, not everyone searches like you did.

Also, just to be safe, a standard button hole saw size is 1 1/8", correct?
The standard is 1 1/8", yes.
Thanks.
Welcome.
