The only file of interest is the one I mentioned, wininit.ini.  If not
present then we need to look for another reason why you are seeing the
restart message.  BTW I somewhat doubt that wininit.exe (an executable
process) looks anything like wininit.bak or the other files you mention
all of which are copies of previously executed wininit.ini files that have
been saved for some reason.  Normally a wininit.ini file is deleted once
its contents has been processed by wininit.exe as the system boots.

Since you don't appear to have a copy of wininit.ini present that might be
stopping you from creating a checkpoint could you please try the
following.
Click Start | Run and enter in the box
SCANREG  /FIX
and click OK.  Note that the system will re-boot so don't have any
applications open when doing this.  I suggest this because the same
message about restarting when trying to create a checkpoint is also
displayed when the registry has been corrupted in some way. scanreg /fix
attempts to repair that damage.

Please try the following quick test:
a) Create a shortcut on your desktop to a file (any file will do).
b) Create a manual checkpoint
Could you create a checkpoint?
c) Delete the shortcut you made at a)
d) Restore your PC to the checkpoint you created.at b)
Was the shortcut restored?  Did you see any error messages?
e) Reboot your PC
Was the checkpoint retained?

If the test fails please post back the full text of any and all error
messages seen and we can try taking it forward from there.