In Italian, the word Contessa means Countess. Looks as if they just chopped the Con- off ... I very much doubt if it's a reliable site!
The different spellings of
Theresa just reflect its history. The Th- ones retain the old spelling, though we no longer pronounce it as we do in words like this and fifth. Most languages don't have the [th] sound at all, so they either keep the spelling but don't pronounce it, or they drop the -h-. It's just for convenience.
It's perfectly possible for names to look the same though they might originate in different languages and mean different things. Not very likely, but it does happen. Same with ordinary words. But I don't think there's any mystery about
Tessa - it started as a shortened form of
Theresa and became an independent name;
Terri or
Terry did the same (for a girl. For a boy,
Terry comes from
Terence!).