Response from Autodesk Support:
I understand that you're having issues with Lathe machine configurations.
Lathe configurations are not approved for use in Inventor CAM, only mill/turn. This is why it works when you add the Y-axis.
The Generic Autodesk Lathe formats you speak of are holdovers prior to the functionality being removed. I remember hearing why they weren't deleted from the library a long time ago, but I don't recall the specific reason. The functionality of lathe configurations will not be added to Inventor CAM in the future, but they will be for Fusion.
TLDR: 2-Axis Lathes do not require a machine configuration and therefore Autodesk removed the functionality.
This is why the generic lathe template works but you cannot modify it. You can still program 2-axis, but they are considered simple and the config is done solely by the post processor.