These are different power connectors.
The connector on the left in the photo is a IEC60 320-1 C15 plug (male) which is compatible with the IEC60 320-1 C16 socket (female). The connector on the right is an IEC60 320-1 C13 plug (male) which is compatible with the IEC60 320-1 C14 socket (female).
So from your description the C13 fits but the C15 doesn't. This makes sense because in addition to the key cutout on top, the plug tends to be a little bigger. According to some, the C15 connector should fit into C14 inlets while the C13 cannot fit into C16 inlets. But this is in no way a universal thing and IEC60 320-1 does not require it. The C15/16 connectors are generally for high temperature (up to 120°C) applications. The C13/14 series top out at 70°C.
If you want universal compatibility, I would stick with C13/C14 connectors. They are much more common and any equipment where the chassis temperature does not exceed 70°C should have a C14 socket not a C16. Here is a chart showing the differences.
Attachment:
IEC C13 vs C16 power connectors.png