To the OP: first, I don't know what I would've done, I'm only human, I make tons of mistakes, and I often agonize over the mistakes I made, things I said, or didn't say... so, I'm not here to criticize. I can say, however, that when the objective is to influence someone to do something differently, certain tactics seem work better than others, and I'd like to share with you what I think works. I very often fail to follow this advice, but when I do manage to follow it, I can very easily tell the difference.
Contrary to what's been suggested above, I think it's great that you are being self-conscious and asking for feedback, that's the only way to learn something.
The first thing about influencing people is that we easily get emotional, often without realizing it, and as we grow more emotional, feel threatened, or otherwise uncomfortable, we stop thinking clearly, we get defensive, stubborn, inflexible, hard to reason with. In any situation, in which you want to influence another person, even in a negotiation, in which you want to gain something, the first thing is to make sure that the person is not feeling threatened, and if you notice signs of discomfort, either in their body language, or in the way they speak, you would be well advised to back off a little, and tackle that problem first.
This might seem counter-intuitive, one would think that the best way to convince a person of something is to crush them with the weight of your irrefutable arguments, but that would only work well if we are interacting with a rational mind. In reality, we are emotional, more than we are rational, and it's the emotions that we need to confront. If we humiliate someone, even if they go silent, admit a defeat, and appear to be convinced, there's a good chance that they will feel intense resentment towards us, and they will spend the rest of the day trying to think of ways to discredit us, and everything that we represent. And even if they do not resent us at first, they will most likely come to resent us later, as they think about what happened, and how we made them feel. They might even deliberately go against our good advice. Come to think of it, this makes plenty of sense. After they have been humiliated, they would now be taking another insult, by admitting that they were wrong, and in way, justifying the way, in which they have been badly mis-treated. It is hard to disassociate rational argument from the feelings that accompanied that argument, so it is best if we make the point we want to make without creating any sort of negative feelings.
Of course, this is easier said than done, but it's also not all black and white...
Indeed, the best way to convince someone would be, to approach them as if you approached a good friend, with whom you want to maintain a great relationship. This is sometimes also called being "soft" on the people, and "firm" on the point you want to make. Unfortunately, very often, we all do the exact opposite: we are "firm" on the people (we are rude, aggressive, or impatient), and "soft" on the point (we do not communicate it clearly), and that tends to lead to very poor outcomes.
Others have already pointed out some of the things that could have hurt the instructor's feelings, and that were counterproductive: offensive language, criticizing in front of a student, talking down, showing attitude, etc.
There are a few more examples of this sort that I'd like point out:
1) Criticizing someone out loud such that they can overhear it, without addressing them directly. Why would we want to whisper behind someone's back, rather than delivering the criticism directly into a person's face? Whatever the answer, it generally implies something negative either about us, or about the person we are criticizing. Come to think of it, this is also creating an asymmetric situation, in which we can state our criticism openly out loud, while the other person is feeling discouraged from responding, because the moment they do, they will be putting themselves out there, and taking the risk of being publicly humiliated. This is not a fair game, and a lot of people, myself included, resent that, and if it's done on purpose, perceive it as hostile. The knee jerk reaction may be to get confrontational ("if you have something to say to me, have the courage to confront me, rather than mumbling behind my back"), and that's definitely not helping you as a way of making introductions.
2) Walking out on someone without a word, shrugging, ignoring them, etc., depending on how it's done, we are basically telling someone that they are not worth talking to. At this point, we are not even criticizing their behavior, we are criticizing and rejecting the person as a whole. I cannot imagine a situation, in which this would be constructive. Sometimes, we do that simply because we feel helpless or tired, but that does not make it an effective strategy for influencing people, and in walking out, we are effectively accepting that we have failed.
3) Confronting the person's dumb remarks only makes sense if what you heard is really what they were trying to say, but that's often not the case. We all often speak nonsense when we get emotional, since we cannot think clearly. Or at least, I can definitely relate. I sometimes get frustrated about what I just said while the words are still coming out of my mouth. We are not robots, we do not mean everything we say verbatim. More often than not, when we say something dumb, what we are really trying to say is "I am feeling frustrated, humiliated, or cornered.". It is important to recognize when that is really the case, and to respond to what's the real message. If you don't, and confront the person about something dumb they said, you're only making things worse, for several reasons. It's hard to admit we said something dumb, and we all feel the need to appear consistent, so people will have an incentive to defend what they've just said, and only bury themselves deeper. The whole conversation will get more uncomfortable, and more irrational. That certainly isn't helping.
The above illustrates a broader point: when arguing with someone, it's important to know at what level the argument needs to be taking place, whether it's about feelings, specific behavior, procedures, principles, etc., or else there's a good chance of talking past each other.
There's a mechanism called a "ladder of inference" that illustrates that, let me give you a somewhat extreme example. You walk past a person, you say "hi", they don't say "hi" back. You think to yourself: they didn't say hi because they think they're better than you. They probably think that because they're Japanese, and you're Swedish, or because... and then you begin to resent the person, because you feel insulted by what you think were the reasons for their behavior, where in reality, the reasons might be innocent. You see where this goes. Sadly, this is the way our brains are wired, and we naturally tend to jump to conclusions all the time. In the midst of a heated argument, or any time we are under stress, this tendency only gets worse.
As pointed out earlier, when trying to deliver critical feedback to someone, we should spend 1% of the time talking, and 99% listening. Listening shows that you respect the other person as a person, and helps them feel more positively about you. It also helps you understand what's in their mind, what were the true reasons for their actions, and to respond appropriately. There's a chance that you could use this opportunity to both learn something from each other, and even make friends in the process. You will feel better, and you will do a favor to the future students, who won't get entangled by a line laid in mid-water...
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