I guess for many people who want to know just enough to do their job, the theoretical side can seem daunting and even unimportant. But I'm firmly of the opinion that we should understand WHY we do something (instead of just saying "because that's the way it's done"). That theory may suddenly become very useful when running up against a less common problem.
Of course, I'm just an electronics nut who's wanted explanations of everything since the age of about 5. Others may see it differently!
The capacitor in the control gear of the pump motor is most likely part of the normal split-phase induction motor. It's used to generate the appropriate phase shift to the auxiliary windings to obtain the rotating field needed to start the rotor turning.
There are some variations in fluoro. light wiring. I don't think I ever had cause to open one up while I was in America, so the others will have to clarify that point.
Most conventional fluoro's in the U.K. have the p.f. correction capacitor just clipped inside the housing and wired directly across hot & neutral at the terminal block. Many cheaper types sold for domestic use don't bother with it; the lower p.f. of one or two fittings is considered inconsequential. (And it saves all of $2 per fitting!)
Capacitors are used widely throughout electronics, but one other area that may be of concern to you is filters in power supplies.
These typically have caps. connected H-N, N-G, and H-G in combination with some series inductors. It's becoming more of a problem in big offices, etc., where with dozens of computers in use the leakage current to ground can be appreciable.
