inductors ~ mass. resists sudden change in current. so current "falls behind"
capacitors ~ springs. resists sudden change in voltage so current "gets ahead" or voltage 'falls behind"
Leading/lagging refers to the amplitude of current relative to voltage waveform. The time/magnitude relationship of the voltage waveform is more or less dictated by the PoCo.
At field technician level, I have to say the math that goes behind it is not useful.
Capacitors get installed as prescribed or according to a guideline established by OEM or engineers. Fan gets 5µF and compressor gets 30µF, based on manufacturer's specs, published guidelines or as directed by others. And as far as what techs need to know, it induces the proper amount of phase shift to produce the rotating field needed to produce starting torque in the right direction.