The Story of Mario

[Slimybug]

Hello.

I was recently looking to doing some research into the Mario games, and was curious about the storyline, or lack thereof. I know the games are notorious for not really having much of a story, or explaining much about the characters. I know there was an unofficial story explanation that someone came up with some time ago, which did the rounds and became very popular. But I want to see if I can't mostly stick to the official stuff.

First of all, where was Mario first named? I see his name is in the Donkey Kong manual, but since the game was origianlly for arcade, without a manual, I wonder how people knew it.

The first game manual that comes with a story is Super Mario Bros, in 1985.

One day the kingdom of the peaceful mushroom people was invaded by the Koopa, a tribe of turtles famous for their black magic. The quiet, peace-loving mushroom people were turned into mere stones, bricks and even field horse-hair plants, and the Mushroom Kingdom fell into ruin.
The only one who can undo the magic spell on the Mushroom People and return them to their normal selves is Princess Toadstool, the daughter of the Mushroom King. Unfortunately, she is presently in the hands of the great Koopa turtle king.
Mario, the hero of the story (maybe) hears about the mushroom people’s plight and sets out on a quest to free the Mushroom Princess from the evil Koopa and restore the fallen kingdom of the Mushroom People.

The Mushroom Kingdom is made up of a number of worlds, and each world is divided into 4 areas. The fourth area of each world ends in a big castle. The Princess, as well as her mushroom retainers, are being held in one of the castles by the turtle tribe. In order to rescue the princess, Mario has to make it to the castle at the end of each world within the given time. Along the way are mountains, pits, sea, turtle soldiers, and a host of traps and riddles.
If you come across mushrooms who have been turned into bricks or made invisible, they reward you by giving you a power boost. With each boost Mario changes into a different, more powerful Mario.

No reference here to where Mario is originally from. I wonder where the first reference to him coming from Brooklyn was?

The second game is largely ignored, but the story from the manual shows Mario living in the Mushroom Kingdom alongside Princess Toadstool and the others, apparently after a happy ending against King Koopa.

Super Mario Land, in 1989, shows Mario traveling to the different land of Sarasaland, to save Princess Daisy.

Then, later in 1989, came The Super Mario Bros. Super Show, which, if it wasn't made clear before, established Mario and Luigi as Brooklyn plumbers who were sucked into the Mushroom Kingdom via a random pipe they were working on. The show apparently picks up after the events of the first game, with the opening narration to the first episode being Me and my brother Luigi had just rescued Princess Toadstool from King Koopa, and were looking for the magic that would set her kingdom free, and get us home to Brooklyn."

Future games feature Bowser continuing to kidnap Princess Toadstool, eventually called Princess Peach.

At some point, Mario was portrayed in commercials with an Italian accent, despite the Brooklyn origins showcased in the show (and the live-action movie). This led to the unofficial storyline stating that the brothers were from Brooklyn, but took their accents from their adoptive parents.

So, I know there's some stuff I'm missing, and that one game even shows Mario and Luigi as having been born in the Mushroom Kingdom originally. Anyone want to fill in the gaps, or let me know where I should look for more? Has there ever been any semblance of the story moving forward?