Long question! Really glad you asked this.
The core loop in pretty much any fighting game can be described in two simple phases -- the "neutral" phase, where both players are freely able to control their characters and play to land that first hit, and the "resolution" phase, where the player that lands that first hit gets to convert that into an advantage.
Depending on the game, the "neutral" can look like footsies (SF games), or navigating around fireball zoning and assists (Marvel/Skullgirls), or full-screen poking with big ass moves and crazy air-dashing and teleporting (Guilty Gear). The fireball/DP stuff still exists in SG, but it's a smaller % of the neutral game when you have assists and air block.
And once you land that first hit, the game might be designed to only give you a little more damage and screen control (say, c.MK xx fireball), or you might get to kill your opponent's first character with a 10 second combo (Marvel/Skullgirls).
I like ST/HDR as a starting point because you need to have decent execution to perform moves consistently in those two games, and because the "resolution" phase is pretty small; you hit someone, you maybe get a small combo, or a knockdown that gives you a positional advantage, and then you play the neutral game again. (As you get better at the game, these things can get more important -- a Ryu in the right spot should be able to shut Honda down pretty badly -- but for newbies, this often isn't as much of a problem.)
In Marvel or Skullgirls, if you give up that first hit, there is a good chance you lost the game. (Seriously -- most Marvel games end up coming down to who lands the first clean hit in my experience.) On one hand, this means that learning the neutral game is incredibly important (one mistake and you might lose!), but because the game allows you to convert one hit into a character kill, it means that you need to learn how to do that before your neutral work will matter. It doesn't matter if you can beat me in the neutral game 80% of the time if you need to land 10 clean hits to kill my character and I only need to land one.
If you want to learn to play Skullgirls, you're gonna have to put some work into practicing combos. If you want to learn ST, you're gonna have to get good at performing reversals. All of these are fundamental fighting game skills! But games will stress them differently.
As far as writing about fighting games in a way that would illuminate these differences: It's something I've thought about, but honestly, I don't think it'd be that useful yet, because IME most would-be FG players don't really have a wide range of choices when it comes to the games they can easily find people to play locally with, and if you're into FGs for the competitive/social aspect, it's better to play a less-than-ideal game like SFV with people than a niche/older game by yourself. For this crowd, community size is important -- it means you're more likely to find quality online/offline matches.
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