I think you're right about the second. I'm not sure, but I have to say that when I first saw your first photo, and I hadn't scrolled down to the second, P. haselbergii was my thought for that too!
Definitely not a Mamm, anyway, says he sticking his neck out....
There's a simple test for a Mamm - are there axils at the bottom of the tubercles? If not, then what you have is almost certainly South American, and probably a Parodia.