Rummy version with a run of seven as last round
Sounds like you've already found the answer: it's a form of contract rummy, just with a different final contract than the one listed on the Wiki entry.

From the contract rummy entry on rummy-games.com :


  The rules detailed below are representative of all Contract Rummy
  games, though there are numerous regional and local variations even
  among games with the same name.


Incidentally, that particular contract (7+3) appears as one of the contracts in Shanghai Rummy (but not the last).