Odd, though, that you can still move left/right if you rename "Move Left" and "Move Right."
If a control does not exist (as in, it was deleted or renamed to something else), the behavior reverts to the default controls. This is so even if you delete all the controls you can still use the default controls. If you rename "Move Left" to "My Control", it reverts to the default control, so if you change which key "My Control" is, it won't affect the behavior.
Basically on startup plugins ask Construct what a key is by its control name, and if it's not found, it'll use a default.