‘what made A experience this as a betrayal’ is the fact that it was. It really is that simple. You could perhaps object that it is strange to experience vicarious betrayal, but since it sounds like the four of you were a team, it isn’t even that. This is a very minor betrayal, but if someone were to even minorly betray my family, for instance, I would automatically feel betrayed myself, and would not trust that person anymore even if the family member doesn’t actually mind what they did.
Analogy time (well, another one), ‘what makes me experience being cold’ can be that I’m not generating enough heat for some personal reason, or it can just be the fact that I am outside in −20 degree weather. If they had experienced betrayal with the person asking for permission to do a move that was better for the group, that would be the former, but this is the latter. Now, it obviously can be both where a person who is bad at generating heat is outside when it is −20 degrees. (This is how what you are saying actually happened works out in this scenario.)
From what I’ve seen of how ‘betrayal’ is used, your definition is incorrect. (As far as I can tell) In general use, going against your agreement with another person is obviously betrayal in the sense of acting against their trust in you and reliance upon you, even if the intent is not bad. This is true even if the results are expected to be good. So far as I know we do not have distinguishing words between ‘betrayal with bad motives’ and ‘betrayal with good motives’.
Another analogy, if a financial advisor embezzled your money because they saw a good opportunity, were right, and actually gave you your capital back along with most (or even all) of the gain before they were caught, that is still embezzling your money, which is a betrayal. Since they showed good intentions by giving it back before being caught, some people would forgive them when it was revealed, but it would still be a betrayal, and other people need not think this is okay even if you personally forgive it. Announcing the course of action instead of asking permission is a huge deal, even if the announcement is before actually doing it.
You can have a relationship where either party is believed to be so attuned to the needs and desires of the other person that they are free to act against the agreement and have it not be betrayal, but that is hardly normal. If your agreement had included, explicitly or through long history, ‘or whatever else you think is best’ then it wouldn’t be a betrayal, but lacking that, it is. Alternately, you could simply announce to the group beforehand that you want people to use their best judgment on what to do rather than follow agreements with you. (This only works if everyone involved remembers that though.) The fact is that people have to rely on agreements and act based upon them, and if they aren’t followed, there is little basis for cooperation with anyone whose interests don’t exactly coincide. As you note, their objection was not to the course of action itself.
The damning part isn’t the fact that they thought there was a new course of action that was better and wanted to do it (very few people object to thinking a new course of action is better if you are willing to follow the agreement assuming the other person doesn’t agree), it was the not asking and the not understanding which both show a lack of trustworthiness and respect for agreements. This need not be a thing that has happened before, or that is considered super likely to occur again for it to be reasonable for another party to state that they hate such things, which one of the things being communicated. One thing objecting here does is tell the person ‘you are not allowed to violate agreements with me without my permission.’
Also, they may be trying to teach the violator, as it is often the case that people try to teach morality, which may be why so much of philosophy is morality discussions. (Though I don’t actually know how big of a factor that is.)
If there had been a reason they couldn’t ask, then it would make more sense to do the seemingly better thing and ask for their approval after the fact. This is often true in emergencies, for instance, but also in times of extreme stress. Your friend wouldn’t feel like it was a betrayal if the other person had instead gone to bathroom and never came back because they got a call that their best friend had just been hit by a car and they didn’t think to tell people before leaving. If, on the other hand, the person acted unable to understand why they should explain themselves later, or that it wouldn’t have been better if they had remembered to do so, that would be bizarre.
I do agree that considering the hypothesis that they may have experienced serious betrayal is useful (it is unfortunately common), which is why I think asking about it was potentially a good idea despite being potentially very awkward to bring up, but I think it is important not to commit to a theory to degrees beyond what is necessary.
I also agree that feeling understood is very important to people. From what I can tell, one of the primary reasons people don’t bother to explain themselves is that they don’t think the other person would understand anyway no matter how much they explained, with the others being that they wouldn’t care or would use it against them.
For something to be a betrayal does not require knowing the intent of the person doing it, and is not necessarily modified if you do. I already brought up the fact that it would be perfectly fine if they had asked permission, it is in the not asking permission to alter the agreed upon course where the betrayal comes in. Saying ‘I will do x’ is not implicitly asking for permission at all, it is a statement of intent, that disregards entirely that there was even an agreement at all.