Well, that's indeed different. Thanks for correction.
I've spent some time thinking about it and I have to say that I would rather put circular reasoning as:
p -> p
as the simplest possible case. Yours also works, but permits whole case to actually be false. I'm trying to make some good example based on what you wrote to have it a bit clearer, just as an exercise. If you happen to have an example as some form of 'word problem' I would like to see it. Before you'll ask: yes, it pains me to admit it but I can be that shitty with English ;/. My recent preparatory IELTS exam made me realised how many moronic mistakes I can make, so that's partially a help request if you happen to have the will and time to do it.