Thanks Nathan for taking the extra time.
Looking at the keyway under magnification it looks more like 1013c.
The 1014 is 8.36mm wide and the keyway in my lock is ~7.3mm.
The key that I showed in my OP fits the top part of the keyway in the lock with the rectangular piece but does not reach the bottom of the keyway. It does fit the smaller lock.

Also as I asked in my OP can you tell me what the purpose is of the rectangular piece in the plug?