As you found, the games themselves appear within the steamapps\common folder. But in order for Steam to know about the game, you also need the related appmanifest file. These are found within steamapps folder. Example appmanifest_440.acf is necessary for Team Fortress 2.
On a related note, if you need to check where else Steam thinks games are installed, within the Steam client navigate to Steam>Settings>Downloads>Steam Library Folders.