Your examples are distinct
Damage rolls, in particular, are completely different from any other check, since they do not use a d20, but rather anywhere from 1d2 to 2d6 (and that’s just for player-race-sized options!) plus various “damage bonuses” that vary from weapon to weapon (non-composite projectile weapons get none, light weapons get half-Strength, one-handed weapons get Strength, two-handed weapons get half-again-Strength, composite projectile weapons get a fixed number, and exceptions and special cases exist for each) and from class to class (rogues might add Sneak Attack damage, rangers may add Favored Enemy bonuses, etc).
But attack rolls are also distinct from any ability checks, whether that be Strength or Dexterity. For one thing, they use Base Attack Bonus and auto-succeed on a nat-20 and auto-fail on a nat-1, and have critical threats and critical hits, and all kinds of other things that apply “attack bonuses,” and ability checks do none of these things.
Skills too, for all they represent almost the same thing as ability checks, just with training added, are not technically ability checks. Bonuses to ability checks wouldn’t apply to skill checks (which is why almost every single bonus to ability checks is actually explicitly a bonus to ability checks and to skill checks using that ability).
Some rolls are subsets or specific types of other types
The big one is Initiative, which is a Dexterity ability check. There probably are other examples. But in every case, this is explicitly noted:
At the start of a battle, each combatant makes an initiative check. An initiative check is a Dexterity check.
(emphasis and emphasis mine)