This is a form of contract rummy called progressive rummy.
(I note that since the contracts you name include ever-increasing ones eventually leading to ones that cannot be met with only seven cards that each deal probably involves an additional card)
Progressive Rummy: Progressive Rummy is played very similarly to Contract Rummy. The main differences are in ...
You describe a variant of a Rummy game. With the following special rules:
2's act as jokers.
Special pick-up-from-discard pile rule: can pick up any number of cards from discard or a single face-down card from the deck.
Target goal of points, similar but not the same as in Pagat:Kalooki
starting hand of 9 cards
Paget:Rummy have a long list of Rummy games, ...