What was the name for the normal poker game?

Poker.

5 card draw.

Texas Hold Em is not a normal poker game, it is a casino poker game.

7 card stud is a much better game.

That is similar to asking what is the normal term for a house. There are ranch, A-frame, cape cod, and other styles, but each are houses, and common.
Poker is poker, and usually in non casino games, dealer calls it.
One could consult the Hoyle book of rules to get a clearer understanding of variations.

Draw or stud poker

In stud poker, you get only the cards you’re dealt initially
I’ve played 5 card draw where you can draw up to 4 cards, but the remaining card held must be an ace. Otherwise, 3 is all you get after you discard 3.