Oh, I thought of another one.
Super Mario Bros. 2 (the USA version) is my favorite of the NES Marios, and I think all the people are ridiculous who say it's not a "real" Mario game just because it started life as Doki Doki Panic.
It is too a real Mario game. Many of the enemies introduced in SMB2 have become recurring in the franchise (including Bob-ombs, Ninji, Pokeys, and Shyguys). It was the first Mario game to let you play as Princess Peach or Toad, and first to really establish the different strengths and weaknesses of the four characters.
Not only is it a "real" Mario game, it's much better than the Japanese version (a.k.a. "The Lost Levels"), which though it innovated in some ways was too much of a rehash of the first SMB.
SMB2 is a great game. One of my favorites.
Super Mario Bros 2 has always been my favorite of the series, as well, and probably my favorite platformer ever. Unlike SMB1 (which is still a great game), it had bright colors, large levels, vertically-scrolling levels (as well as the ability to backtrack left), worlds that looked and played completely differently from one another (ex. lush levels, desert levels, ice levels), four very characters to choose from (and you could switch between them between levels), four different ways to play the game, more diverse enemies (because many were specific to their worlds, like the cactus enemies on the desert levels) and more. The replayability was immense. SMB1 had the same experience from the first to the last level and, if you played it with Luigi in 2-player mode, literally nothing changed other than the fact that you were wearing green overalls instead of red.
Even though it's so different from SMB1, what makes SMB2 "true Mario," IMO, is the fact that, after its release, so much of what was established in SMB2 became part of the franchise. It wasn't just abandoned. Later games in the series incorporated larger, vertically scrolling worlds, worlds with specific themes, more characters to play, SMB2 enemies and so on. SMB2 added a lot to the evolution of the franchise. In fact, when you consider spin-off series like Mario Kart and Mario Party, which are built around the concept of a collection of playable characters, all with unique personalities, traits and play styles, you can't help but think back to SMB2 introducing that concept. SMB2 really had a strong impact on the franchise, IMO, and it happened because of a mistake. If The Lost Levels had been SMB2 outside of Japan, as it was planned to be, SMB might've never grown past infancy to become what it has, which it has been because of continual injections of creativity (which The Lost Levels didn't have and SMB2 did).