A check implies action.

The point I'm arguing, that you don't seem to be understanding, is that the word "check" implies that something must be done, but the word "passive" means that nothing must be done.

But the implication of the word "check" creates a lot of unnecessary confusion about how the system is actually supposed to work.

You're really the one missing the point here. "Check" does not imply action. You, and some new players and DMs, think it does. And maybe from the very beginning they should have used a different word. But this is what we've got.

A check does not imply action. A check is a mechanic used, here in the actual real world, to adjudicate something that is uncertain in the game world. A check is not something that actually happens in the game world. A check determines the outcome of something that happens in the game world, when the outcome is uncertain.

This is crucial to this entire argument. It's an important point for players to understand as well, if you want them to be immersed in the game. The check is not the thing. The description of what happens in the game is the thing.

There's lots of life that you can breathe into your game if you stop thinking about passive skills as checks to be made and instead something that's always present. Little details you can throw out, like the weak party member walking into a door because it didn't automatically open for him (and everyone has a good laugh), or describing the difficulty of lighting a campfire because the low-Wis character volunteered to gather firewood and came back with a bunch of green wood and wet grass. It gets the players more invested in their characters and the skills they chose to see them actually play out in the game. To see the difference between the urban fighter and the rural ranger in the small details of things like gathering firewood.

You know, i said it before and I'll just reiterate it here. This is part of DMing 101. It has nothing to do with this discussion, whatsoever. You don't need to see every single thing that happens in the game as a check to understand that your PCs are not identical copies of each other and to describe their differences as they do stuff. I don't know why you're throwing it out there as some type of insight that will be unlocked if someone sees things your way.