Here's what I did to figure out how fights happen and how to predict who will win: I watched a ton of YouTube videos showing real, actual fights that happen in real life. Others have done a similar thing.
The main thing I walked away with after watching many dozens of street fight videos is this: The one who loses the fight is usually the one who signals he doesn't want to fight. You can usually predict what will happen within the first few seconds of the interaction.
What does it mean to signal that you don't want to fight?
Typically, street fights go down like this. The aggressor gets really upset about something some other guy did and quickly walks up to him and gets within punching range. The aggressor then stares straight at the other guy's face menacingly, pushes out his chest, hangs his arms down at his sides and clenches his fists while raising his voice in anger. The aggressor usually does micro aggressions which twitch his shoulder or forearms in a manner that indicates he's starting to punch, and he's watching the defender's reaction to it to gauge whether the defender will fight back. Sometimes if the aggressor gets closer, he begins shoving the defender's chest, pushing him back. This is pretty common. He's again trying to gauge whether the defender will fight back or not.
The defender then has two options. He can back down or stand up to the aggressor. If he backs down, he typically steps back, lowers and calms his voice, and says he doesn't want to fight or asks him what his problem is. His eyes usually stare at the guy, but occasionally you can see he's looking to the side to indicate that he doesn't want to be there and wants to leave. That's pretty typical of what someone looks like backing down and not wanting to fight.
Sometimes it's just that the defender stands there looking kind of weak and averts his eyes from the aggressor. His body just kind of turns to jello, very weak. He's making it very obvious with his body language that he won't fight back, and he's hoping this will be accepted by the aggressor and that the aggressor will take pity on him and leave him alone. He's putting himself at the mercy of the aggressor.
What it looks like for the defender to stand up to the aggressor is for the defender to posture up, stand his ground, and stare right into the eyes of the aggressor. He doesn't take his eyes away. He won't occasionally look to the side. He just keeps staring straight at the aggressor. He might even take a step towards the aggressor instead of stepping back. He often looks relaxed. This says to the aggressor that if he attempts to fight, the defender is ready for it and will fight back. Having a relaxed posture also indicates that he's got the skill to back it up.
Who wins the fight?
The one who wins is the one who punches first, typically. If the aggressor walks right up to the defender and basically talks trash to him, starts shoving him back repeatedly, all while not throwing any actual punches, and the defender is looking like he's ready to fight, then the defender might throw the first punch and win. This is because the defender realizes that the aggressor doesn't really want to fight. You see, if the aggressor hasn't actually thrown any punches after a while of just going off verbally on the defender and maybe even shoving him back repeatedly, then that is a signal to the defender that the aggressor isn't really going to attack. The defender realizes that he has an opportunity to throw the first punch and beat the aggressor, so he does.
On the other hand, if the aggressor goes up to the defender and realizes that the defender is weak and doesn't want to fight, the aggressor may throw the first punch fairly soon in the interaction. That's because the aggressor knows the other guy isn't going to fight back, and the aggressor wants to fight.
So from a self-defense perspective, it's important that you signal readiness to fight to your opponent. That means you don't look away from your opponent. You don't try to back away. You do need to posture up by pushing out your chest, pulling your shoulders up, tucking your chin, and getting into a stance that's ready to punch and ready to sprawl. Get your arms up and ready to guard your head from a strike. You can get your hands up without getting into a boxing punching position, too, so that your opponent doesn't take it as an immediate threat.
It's then up to you as a defender whether or not you're going to throw the first punch. Legally, that could be a problem for you. But just speaking for myself, if it's clear that the aggressor isn't ready to fight at that moment but may still attack me soon enough, I might use that opportunity to throw the first punch. That's because I know that if it connects, generally the odds of me winning and surviving go up a lot. Whereas, if I wait and let him attack me, my chances of survival go down a lot. It all depends on whether or not you think he's going to attack for real. It's not usually obvious, though.
As for what you're doing with your eyes: Think of it like this. If you're not looking at his eyes, you can't read what he's going to do. You're a sitting duck. You're just hoping he's seeing that you're not interested in fighting, calms down, gets bored, and walks away. Yeah, he's yelling at you and maybe shoving you, but you figure that's okay so long as he doesn't start punching you. But that's generally not something you can rely on. Instead, they often will take that opportunity to attack you, because you're indicating that you're weak and won't fight back.
So you need to stare at his eyes. That's very important from a self-defense perspective. It's going to signal that you're not weak, and you're ready to fight. It's also going to allow you to read him and give you a heads up before he makes a move. This is a proven concept in almost all martial arts.
Now, many martial arts will teach you that when someone comes up to you aggressively, you should immediately take a step back, get into a stance, put arms up, palms facing out, and stare at the guy while saying to him that you don't want to fight. This is for two purposes. First, it attempts to deescalate. Fights are bad. Even when you win, you might get hurt. You might even die. So it's best not to fight if it can be avoided. And so deescalation is a pretty good idea. Second, while saying you don't want to fight, you're actually in a position to be able to defend yourself.
Legally, this is the best approach. You're waiting for the aggressor to attack, but you're ready to defend if so. You're relying on your martial arts skill to be able to maintain the distance, read your opponent, and react fast enough to avoid his attack while also allowing you to counter-attack.
That is a popular martial arts theory. The theory isn't necessarily wrong, but it often fails.
It fails for two reasons. First, it overestimates your skill and ability to see an attack coming and be able to react to it quick enough and decisively enough. You think you have enough distance between you two to be able to see anything coming, but it's often not enough. You think you can block a very powerful punch coming, but you may only deflect some of its force before getting hit. Even a grazing hit to your jaw can leave you nearly knocked out and much weaker, unable to fight back. And even if you managed to defend against that strike, you don't know what to do next. Some people just freeze, because this is something they're just not trained to do. It's real life, not play sparring. They don't know what to do. The other guy, however, doesn't care. He just keeps coming. You just bought yourself a few extra seconds before lights out for you.
Second, what is often left out of martial arts training is that you shouldn't actually not want to fight. You're taught to deescalate and indicate to the other person that you don't want to fight. What is left out of martial arts training, in my experience, is that you should just look like you don't want to fight. Inside your head, however, you should have already thrown a switch that puts you in a fighting mindset, so you're ready to fight right then and there.
You actually have to want to fight, and you're just holding yourself back for the right opportunity or until he backs down. If you don't want to fight, you're going to lose. This is an aspect of martial arts training that is often not taught or explained, to the detriment of their students.
In conclusion, staring at your opponent is an important aspect of self-defense. It signals your readiness to fight and makes it less likely you'll be attacked. It also allows you to read what your opponent is going to do before it happens, which is vital for giving you enough time to react and defend yourself. Looking away indicates that you don't want to fight, and your opponent can use that as an invitation to attack, since they know you won't fight back. Who wins in real street fights can usually be predicted by looking at this and other elements of body language within the first few seconds of the interaction.
EDIT: I went back to read the forum in the question's link. They were talking about situations which didn't involve an immediate fight. You're maybe sitting at a bar and someone is staring at you from across the bar. Then it is acceptable to break eye contact. You're not in an encounter just yet. You don't want to provoke him. He's maybe looking for any reason to go over and pick a fight with you. So you just ignore it, look away, go back to what you're doing. If he then wants to come right up to you and start a fight, that's when what I wrote applies. Of course, there are a number of verbal deescalation methods to use before that even starts. But you have to do it before he postures up. You can get a hint of at the following link:
https://youtu.be/-4UEgtt4ZPM?si=1PR0shejEmyMtApK
Hope that helps.