Are these all the same plant?

No, they aren't. Cisanthe is a genus. From what I've read, the genus Cistanthe was created for several species previously classified in Calandrinia. It seems to be one of those botanical puzzles we often encounter. Similar but separate. Portulaca is in a different family of plants.

So are the genus Cisanthe, the same as the genus Cistanthe with a "t" ?

Sorry for the typo, Caroline, the correct spelling is Cistanthe (pussypaws).

But there are plants called Cisanthe without the "t"?

I don't think so. I think you sometimes see that but think it is a misspelling.

Thanks, no wonder this one is confusing!