That game was called "Doki Doki Panic", and it's history is slightly more interesting than you mentioned.

After publishing SMB1, Miyamoto considered designing a game based around vertical scrolling as it's successor. Of course, it was developed as a much different game with different characters, becoming a game called "Doki Doki Panic". Meanwhile, a secondary team at Nintendo began work on a more direct sequel to SMB; this is the Japanese SMB2.

Nintendo of America saw the Japanese SMB2 and a number of people at the branch hated it for a few reasons. First, yes, it was bloody hard, compared to the first game's really polished difficulty curve. But it was also a mission pack disguised as a sequel. It's like going from Call of Duty: Modern Warfare 2 to Call of Duty: Modern Warfare 3. The most radical visual change in the game was an updated tileset - which, in my opinion, looked worse. It has that "early NES feel" where the artist draws metatiles without considering what they look like next to each other, which is amateurish.

Japanese SMB2 had some new features, mostly to hinder the player's progress. Poison mushrooms, long chasms that have to be navigated with gusts of wind, and a few other big difficulty spikes. But ultimately it looked and played the same as SMB1. I personally consider it an inferior sequel because of it's blatant unoriginality, and NoA agreed, so they took Doki Doki Panic, changed the sprites of the four hero characters to be Mario characters, and thus we got International SMB2.

The home branch of Nintendo was not made unaware of the switch. In fact, they released the modified international release in Japan as "Super Mario Bros 2 USA", which would indicate that Nintendo proper approved of the change. And Nintendo of America did wind up releasing the Japanese SMB2 - in the form of additional content for future SMB1 releases, in both All-Stars (Super Mario Bros: The Lost Levels) and the GBC port (Super Mario Bros For Super Players).

Miyamoto himself has commented that his favorite Mario game is International SMB2 / Doki Doki Panic. And many enemies from Doki Doki Panic / International SMB2 have been inserted into the Mario canon of enemies - most notably, Shyguy, who reappears as early as Yoshi's Island (if not earlier).

SMB3 did not suffer from this problem, primarily because development was headed by Miyamoto, and thus it was a much superior sequel to Japanese SMB2, and NoA didn't feel the need to confuse us about their parent company's games. There's a bigger variety of enemies, the art is much more aesthetically cohesive, the level design is more varied and interesting, and all the other things you would expect from a sequel.