What happens if you install a game to an external drive and then remove it? Does the game just not work if you try to start it? Does it get "uninstalled" according to Steam? Does Steam have serious problems? What if you plug it back in later? Does it work easily, or does it take a lot of work?
Installing to the same directory when it offers & requires installation detects the game and the game will work.
Answer in Greater Detail
I installed VVVVVV on an external hard drive. The game played fine. I unplugged the drive and got this error:
I plugged the external drive in again and the game worked fine again.
Then, I unplugged the external drive and restarted Steam. The game was listed as uninstalled. After plugging in the external drive, the game remained as uninstalled, and launching it would prompt installation (in addition to creating a new Steam library on the external drive). Installing to the same directory at this point immediately installs the game and it starts working again.
Restarting Steam also detected the game again and it played fine.
I like to answer every question that I can reasonably test exactly in that fashion; by reasonably testing them. So let the science begin.
I installed Super Meat Boy on a flash drive for this test. Normally, you would probably use an external hard drive, but the principle is the same. The game runs well, and I played a couple of levels, to see if saves would be affected.
I safely removed the drive and attempted to run Super Meat Boy again, and it gave me the following popup:
It still gives me the option to play by default though, and upon plugging the drive back in, I can play without any issue, saves are retained, and stored on the drive.
I didn't install my games to a removable drive, but I did install them onto another partition. Instead of unmounting the partition though I simply renamed the folder in that partition which I designated as my Steam library. What happens then is that the folder no longer appear in my list of Steam libraries, and the games themselves appear to not be installed.
Restoring the original folder name is enough to have Steam recognize my games again as long as I don't modify any files inside it unnecessarily.
steam just can't find the game when it attempts to start it up. when you click on the game in steam it's like a shortcut and if the shortcut is to a place that isn't on the computer at the time of pressing it it can't open it and steam will tell you just that.