Secret doors are really just very elaborately concealed doors. Somthing blocks the observer from immediately noticing the door. But how does this separate it from the door that is hidden behind say a curtain or tapestry?
It seems like the real difference is that one kind does not have it's camouflage built in while the other does. The door to the Lonely Mountain, the gates of Moria, the bookcase that swings out to reveal a hidden passage, the fireplace that rotates to do the same, the door that is designed to look like part of the wall etc. etc. on and on are all just concealed doors that exist on the cleverness of the concealment spectrum. And so the classic roll to spot a secret door in my mind should be combined with the roll to detect a concealed one.
And this could work quite well in any system. For example with "Set Design" method you can establish a clue that exposes the door when someone investigates an area in the right way. The roll to notice is the roll to notice this clue right away - but with the clue still discoverable through careful investigation. This streamlines the flow of game play a bit - rather than being an "only chance" roll the roll to detect becomes something that speeds up the game.
But what is interesting about all this is that there is still space for the "secret" door. That is - the thing that makes a door secret is not that it is concealed (or not), but rather that the means to open the door itself is secret. And this bears out in the trope- the doors to both Erebor and Moria are known to the protagonists. But in both cases how they opened was the deeper mystery. For Erebor the big secret was the location of the key hole. For Moria it was the word that opened the gates. With the bookshelf it is finding out what book to pull, or where the catch is on the fireplace.
So from now on in my games:
A door designated as hidden is one where someone has taken effort or nature has accidently concealed a door. A door designated as secret has some riddle about its opening rather than just being hidden.