You're all correct!
When Trinity was first a 'thing', it was a bundle of vanilla, plus both expansions. You would get a registraton for each. Once the disc version came around, it changed to being one registration. With Trinity, you get the exact same content, you're still able to play, for instance, Entrenchment without Diplomacy, and any multiplayer action will be compatible with the non-Trinity versions of the game. When there's an update for one, there will be a corresponding update for the other. It's the same files, even, except for stuff like the launcher.
Hope that clears things up, and happy gaming!