Is there such a poker game such that it is played and dealt exactly like Texas Hold’Em, except you have to use your two down cards to make your hand?
Well, there’s Omaha. In Omaha, though, you’re dealt four cards, and must use exactly two of those four cards along with three of the five community cards. Other than that, it’s played and dealt exactly like Texas Hold 'em.* So, it’s not exactly like Texas hold 'em, but close. Is that what you’re thinking of?
*ignoring the fact that Omaha, these days, is often played hi/lo.
There’s also a game called “Pineapple” where you’re dealt three cards, you see the flop, discard ONE. Then see the turn and the river.
In that game I think you have to use both of your hole cards.
I’ve never played the game the OP describes, though.
In the ‘Crazy Pineapple’ I played in Dealer’s Choice games in the student poker club you didn’t have to use both your hole cards. It’s a bit of a novelty game though and I’ve never seen it spread online, so the rules probably vary a bit from place to place.
UltimateBet has Crazy Pineapple, and they don’t have the ‘you must use both your hole card’ rule (in fact, in their page on the rules of the game, they go out of their way to specifically say this isn’t the case). But, it’s the only place I’ve ever played it, and as you say, the rules may vary at other places.
I wasn’t thininking of a particular game, I was just wondering if this variation which I described might exist outside my mind. Doesn’t sound like it.