I have Conquerors installed (Steam), but some people I'm trying to play with only have Age of Kings. I managed to make Conquerors work fine via voogley, the problem is Kings doesn't.

There is no error screen, the game simply doesn't open after clicking the Launch button. I tried opening it with "Start at Single Player Menu" enabled, still the game doesn't launch.

Game Diagnostics says everything was found and successful. Before that I had to download the empires2.exe separately'cause it was missing.

I tried uninstalling and re-installing AoE2 and Voogley, still nothing. What is missing?

aoeHD from steam does not have separated aok from aoc and compatible patch makes it works for aoc only. You need CD version of the game to play aok alone.

So if my version is from steam then I can only play aoc through voobly?

correct