These features are only an issue in competitive games, because without competition, there is nothing to come back from. In single player games of you vs. the horde, even if you are granted a bonus for being in a dire situation (such as the way limit breaks work in Final Fantasy 8), it's not really a comeback. It's a dramatic victory, which is fun, but not a comeback because you were never really against anyone in the first place.
As for a definition: a comeback mechanic is a purposeful advantage given to the losing player to try to keep the game more even. I know, it sounds like a bad idea already, but game designers are still including these in games today. The main reason is to try to recreate those dramatic victory moments I mentioned as often as possible. I even admitted above that those moments are fun, but they're not fun to everyone. They are fun for the winner, and they are fun for the audience, but they stand out as a huge sore spot for the loser (or, in the case of comeback mechanics, the victim).
After all, the loser worked hard to get in an advantageous position, and then just had the carpet pulled out from under him because his opponent used the dreaded comeback mechanic. The loser at no time had the option of using the comeback mechanic, because he was winning. As a result, comeback mechanics introduce uneven footing into the competitive environment, an environment completely reliant on every player having even footing.
In games where comeback mechanics exist, the metagame is built around the comeback mechanic. Anyone who overlooks the comeback mechanic when designing their strategy is fated to lose. If you try to win, you will lose. You have to try to almost win in order to actually win. It's twisted and unintuitive, and you're probably having a hard time understanding what I'm trying to say. So let's look at two games that have comeback mechanics and how the strategy has evolved within those games to see what I'm talking about.
Marvel vs. Capcom 3 is a game that was long-awaited after the success of Marvel vs. Capcom 2. Without getting too much into what made MvC2 great (which could be an entire series of articles on its own), part of it involved the absence of comeback mechanics. The players were on equal footing at the start of the game and at the end of the game. Everything that happened was completely reliant on player skill. If two of your characters had died and none of your opponent's characters had died, you were at a disadvantage. That's the way competitive games work; most of the game is the two sides vying for just a little bit of an advantage. If you win the tiny battles over and over, it becomes a significant advantage, and that's when you can go for the finisher.
It means that dramatic comebacks aren't common, but it doesn't mean they're impossible, as evidenced by this classic:
MvC3, however, made comebacks the norm. Keeping an advantage going is actually more difficult than coming back from two characters down. The comeback mechanic, in this case, is called X-Factor. X-Factor is available to both sides from the start of the match, and can only be used once per match. It gives you a boost in speed and damage and prevents chip damage. If that was everything to know about X-Factor, it would be a fine mechanic. The thing that makes it a comeback mechanic is that it gets significantly stronger and lasts significantly longer the fewer characters you have left. If you only have one character left, your X-Factor is so strong that landing any hit generally leads to a character-killing combo. It is completely normal to see a lone X-Factor kill 2 or 3 characters in a very short amount of time. All the positioning and strategy from the part of the battle with equal footing is completely obliterated by one broken mechanic.
"Oh, but why doesn't the opponent just counter with his own X-Factor?" I hear you asking. But it's not a viable counter. The one-character X-Factor lasts longer than your own multi-character X-Factor, and your opponent is still faster and stronger than you (just not by quite as much).
The metagame has evolved to always have a character who can take advantage of the highest level of X-Factor positioned last on your team, and to ensure that he is the last one standing at all costs. Essentially, the game has been reduced from the 3v3 tag battles of MvC2 down to a fancy song and dance with 2 opening acts to the main event. It's a blast to watch, don't get me wrong, but it's a broken mechanic ruining an otherwise fun game to play.
Mario Kart is another game with a comeback mechanics, and actually employs two of them. One is a little bit more subtle: whoever is in first place goes just a little bit slower than everyone else. It's hard to pull out into a huge lead, and it's easy to catch back up to first to make the game closer. So we already have a pretty annoying disadvantage to being in the lead, but that really doesn't even scratch the surface. The real danger to winning is blue shells. These items are dedicated comeback machines, and ensure that whoever happens to be in first place now is lucky if he is still in first place at the end of the race. When you use a blue shell, the first place player is knocked out of first place. It's that simple.
The blue shell specifically targets the leader and launches him into the sky, wasting a good few seconds of his time and forcing him to accelerate again from a dead stop. It specifically seeks out the person who has been performing the best so far and punishes him for it. The blue shell has encouraged such ridiculous strategies as specifically riding in second place for the entire race and taking the lead at the very end. That's not fun. In a competitive game, you should be trying to win from start to finish.
At the very core, competitive games should be completely reliant on the skill of the participants. If you are able to pull ahead in the early game, you should have an advantage to work with for the rest of the game. If you make a mistake in a competitive game, you should be punished for it. Comeback mechanics reward mistakes, sloppy play, and drama over skill.
If you're thinking of making a competitive game, please leave out the comeback mechanics. They are primarily frustrating and add a layer of stupid strategy on top of the layers of real strategy that you should be promoting.