Abilities that trigger when combat damage is dealt will only trigger when a creature in combat assigns and deals damage in the combat damage step. It is possible for a creature, like Prodigal Pyromancer to deal damage in the combat phase that is not combat damage.

There are also abilities like the one granted to creatures you control by Arm with Aether that do not care if the damage being dealt is combat damage or not. This ability will trigger whenever the creature with the ability deals damage.

Finally, it is important to point out that that some abilities like Wither and Infect will alter how a creature deals damage, but it does still count as creature dealing damage.