Classic just refers to "any difficulty that's not Ultra Nightmare". So you can run it on the lowest (I'm Too Young to Die) or you could go up to Ultra Violence or Nightmare.
The only difference between difficulties is damage done to you, so ITYTD is the easiest, obviously. As for Nightmare vs. Ultra Nightmare, besides the obvious "you can't die on UN", you also can't reload checkpoints which makes a few skips and such a little different.
Hope that answered your question. :)
First thing I'd recommend is joining the Discord (link should be on the left sidebar). There's tons of resources there, and if you're still struggling a bit then you can ask for help. If there's something specific you want help with people are always down to help, but if it's just getting started, I'd recommend just doing a run or two on your own. If you've watched a run or two previously and want to incorporate some strats, go for it, but otherwise just get a feel for the game a bit so that you can get a better understanding of "what" is being skipped, or "why" we are skipping something in certain ways.
Lastly good luck and have fun, of course. :P
For the record, using the SSG Meathook doesn't allow you to go out of the map and "only see the skybox". It doesn't usually give that much height, not without pairing it with something like bumping into a ramp or something, and even then the act of going out of bounds (such as "only seeing the skybox") is also banned. That said, the other tricks that use Meathook do skip large sections of levels, so if you believe the act of skipping those sections is glitchy enough I can definitely understand that view point! Just want to clarify that we don't simply replace "slope boosts" with "meathooks", as that would definitely defeat the point of NMG.
And definitely add your times to NMG or any%! NMG runs were initially submitted to any% before getting their own category, so it's definitely the best way to do it. I wish you the best of luck in the runs, I'm excited to see the routing around certain fights that we normally don't get to see. :)
Diablo, if you really want to run any% glitchless and don't feel No Major Glitches is "glitchless" enough for you (to me the only real difference is its name, and I do dislike how that confuses people from understanding it's basically the glitchless category to no fault of their own), then just run any% glitchless. No Major Glitches wasn't established on speedrun.com and THEN people ran it, it's the opposite. People run the categories they want, they garner interest in the community (such as the speedrunning discord), and then when enough people are running it (usually it only takes like 3 consistent runners) it's considered for having an actual leaderboard, either on the main boards or as a category extension.
If you want any of the categories to become a thing (NG+ or any% glitchless) then just run it and have fun. You can even make the rules for NG+ yourself, and if people seem interested work on the ruleset with them to make sure everyone is enjoying it as much as possible. Worse comes to worst, you have fun playing the game the way you want to. But the category won't "just appear," people have to run the category before it can "become official," which to be honest I hate that term because any type of run you do should be given the same merit as something like normal any% or NMG.
Just enjoy running the game and don't worry about the bureaucracy of it all, and it'll all fall into place.