Above is the basic semantics, and it can be further optimized, such as by extending battery power for the second stage to use it for circularization, inserting a single purpose satellite-like pusher device above the responsibility boundary at the top of second stage and payload satellite and calling it the third stage, or just adding actual third stage above second stage, etc.
Adding more and more stages improve performance per Konstantin Tsiolkovsky's rocket equation, but it'll add risk factors and also obviously add more dead weights in electronics and engines and support equipment, so 2-3 stages is usually the good balance between performance and risk/costs. You can have as many stages as you want if you think you can handle it.
I think that only applies to liquid fueled rockets. At a minimum the Scout family of rockets (USA) and the Lambda 4s (Japan) both use 4 stages. That is what I was trying to say above. The sibling comment also points out that you at least need a third stage to circularize the orbit, since you can't re-light solid rocket motors. I'm not exactly clear on why its 4 and not 3, but that seems to be the standard with solid fueled rockets.
(ISAS side of Japanese space programs is chock full of political BS, everything from the pencil rocket that continued on from IJN rocket researches to the "unguided" L-4S to the LUNAR-A probe with diameter of approximately 152-155mm to one-man laptop launchable Epsilon LV concept focusing on "civilian low-cost rapid launch demands")