The reason I wanted to take out the "going in combat phrase" was b/c it was somewhat confusing. "Going in combat" implied the champion wasn't already in combat, but to even proc or stack Conqueror, the champion has to be in combat for 4 seconds. I would prefer changing it to "staying in combat" or removing it all together.
As for the "attacking" keyword I used, you have a point there, it's pretty inaccurate. I forgot that "attacking" meant basic attacking, not just dealing damage. The client uses the term "damaging," and that would be a better keyword to use.
As for the concern that not all damaging abilities put a champion into combat, perhaps we can write Shaco's Hallucination clone as a special case. I noticed on the Shaco page: it says that Shaco's clone is unique in that it is one of the few pets that doesn't put the owner into combat when it attacks ("Shaco will not go in combat if the clone damages an unit, unlike other decoys, and because of this he's unable to stack or refresh Conqueror rune Conqueror. ") As far as I can tell, it looks like Shaco's clone is the only instance of dealing damage but not going into combat.
there's a difference between remaining in combat and entering combat. After you enter in combat you remain in combat for 5 seconds. This can be refreshed by re-entering in combat.
Damaging is also not true because there're abilities that deal no damage (i.e walls, bard r, placing zilean's q) but trigger enter in combat effects (i.e conqueror, sudden impact, elixir of sorcery).
Shaco doesn't trigger any of these because he doesn't go in combat when he uses his clone while Annie or the others do.
Saying that conqueror refreshes when entering in combat is what i found to be the only correct option.