There actually aren’t that many different stage counts, but the variation comes because different missions have different delta-v requirements, and different propulsion technologies yield different amounts of delta-v per stage.
The definition of “half stage” is not totally clear. It’s mostly used to refer to the Atlas and Atlas II strategy of dropping engines but not tankage; it’s also sometimes used to describe the very common use of boosters that ignite concurrently with a first stage but which have a significantly shorter burn time, and are dropped before the first stage burns out.
The size of the stage doesn’t make it a half-stage. Upper stages are always smaller than lower stages, typically at a mass ratio of around 1:4 or 1:5. If you have two stages plus a “small kick stage” you have three stages, period.
Saturn V used two stages (and a little bit of the third) to reach Earth orbit, and needed the third to go to the moon. The usual payload of the Saturn V was Apollo, which consisted of two spacecraft: the one-stage CSM and the two-stage LM. The complete Saturn-Apollo stack therefore included six stages, but Saturn V should be thought of as a three stage rocket.
“Two-half” stages and half-upper stages as proposed in recent questions don’t actually exist.
Modern liquid fueled launchers designed to send payloads to LEO or to geosynchronous transfer orbit generally have two stages. Sometimes boosters are added to the first stage; if the boosters and core both ignite at liftoff, it’s debatable whether those count as an extra stage or as part of the first stage. If the core doesn’t start at ground level ignition, the boosters are the first stage, the core second stage, as in Titan III.
Launchers that rely mainly on solid fuel stages, and some older liquid fueled rockets, have too low specific impulse and/or too heavy stage dry mass to reach Earth orbit using only two stages, so they have three or even four stages.