If you play the GCN or PS2 MM Collection version of MM6 it autosaves your progress when you finish a stage, either by death or a boss defeat.
The beauty of having this game on the Wii Virtual Console is that you can pause via the home button, exit the game, come back later, start the game, and resume play. Unfortunately I think we have bad timing for such a release, with the focus being on 3DS and WiiU downloadable games, now.
Well now, it seems that you said the magic word... SAVED. My point again, and forgetting about battery. If you start a game, make progress, turn your game off, then at some point later turn the game back on and are able to proceed from the point of last progression, your info was SAVED. Whether by pressing the word SAVE or inputting a password or code, your game was SAVED.
Now just like Frank Farmer in The Bodyguard, "I don't want to talk about this again." :D
Not sure why this is so protracted. Anyway, with a battery, memory card, etc, there is a distinct save file. The actual data for that current game playthrough is being recorded. You can't get ahead of yourself with lucky guesswork or get things not possible in the regular game or earlier in the game (without save state hacking, but requiring external programs to do that, game genie, etc means within the confines of the game it's not possible). The save file remains active for as long as the battery has power, the data is intact on the memory card, etc. Many NES, Game Boy save files can last 20+ years. Save files are the current playthrough being remembered.
The reason why the older, lower-bit systems had save files last so much longer has something to do with the voltage, the power draw on the battery. This is why batteries for later systems went first despite being newer. Add in an internal clock, something active constantly, and the battery drains real quick. That is how many Pokemon Gold/Silver games (2000) had their batteries go defunct before NES, Game Boy, and SNES games from 1987-96 for example, had their batteries go (many of those are still active. As of 1 year ago, a dozen GB & SNES games I had from 1990-96 still had their save files intact. All but 1. That 1 was Metroid II, which I suspect something broke in the cart, got corrupted, or the battery leaked and damaged the board because the game itself didn't work anymore).
By contrast, passwords are recreating the current game playthrough. Notice some passwords don't record everything. You can see this in some games where lives, energy tanks, etc might get reset and have to be reobtained with each continuation of the game via password. With passwords, if the code is cracked (which can be done through hacking or old fashioned guesswork, compiling data points and figuring things out), one can get things impossible to get at an earlier stage of the game or a really high number of lives, energy tanks, items, etc or sometimes things impossible to get otherwise (take The Guardian Legend. You can give yourself multiple keys, giving access to the whole game, right at the very beginning via passwords). Thus, passwords are not saving data but allowing the user to 'restore' a playthrough by giving them the password (which is just a representation of the data that would be recorded in a save file). Similar yet different. Save files are the real thing while a password continuation is a similacrum of the previous halted playthrough. Even if you get down to the raw foundation of it being hexadecimal values being 'remebered' vs. 'recreated', there is a difference because one records everything, one doesn't necessarily record everything.
So yeah, it comes down to one being the original, one being recreated, one being unmodifiable within the game's tools, one being modifiable within the game's tools allowing for otherwise impossible combinations of items/stats at certain times.