Dear Healers who don't DPS

[HyoMinPark]

That assumes that I know which DoTs go with which classes...

Considering SB got rid of a lot of DoTs and consolidated others, I'll make this simple for you. Here are all the DoTs healers have at level 70 and their durations::
WHM: Aero II, Aero III (you should know these)
AST: Combust II (blue icon with darker blue dots; 30 seconds)
SCH: Miasma (purple/white icon; 20-ish seconds), Bio II (pink/purple icon; 30 seconds); Miasma II (purple/black icon; think the duration may be 15-ish seconds but would need to double check)

If you're in 4-man content, I fail to see how it's a big deal not knowing what other healer job DoTs look like. You should be concerned about tossing your own DoTs up there as a healer. It's not like any of them overwrite (aside from resistance down debuffs but those aren't really DoTs). If you have to give out advice to another healer and you are not healing, then familiarize yourself with the different DoTs. Again, SB got rid of a lot of them, so it's not that hard to identify them. They all look different. Same thing with DPS. You can hover over the DoTs to see what they're called to help familiarize yourself with them.

Thank you captain obvious for telling me a bunch of stuff about my own class that I already know.

Please stop assuming I'm the noob here.

You're making healer DPS WAY more complicated that it needs to be. A tank is doing a large pull? As a WHM, you have the perfect mitigation for this: HOLY. Big damage, and a stun that lasts for at least 3 casts before the immunity sets in. The bigger the pull, the better because it's more damage you're dishing out. Thin Air + Presence of Mind + Holy spam is god-tier healer DPS. When I Holy spam, I keep the tank targeted, so that I can Tetra or Bene him if need be. Perhaps try that next time.