A plausible way to cash out "X is being arrogant" is "X is claiming more status than I think they have." That means it's a function of my beliefs about how much status I think X has (implicitly, relative to some community), not just a function of X's behavior in isolation.

should we be dividing arrogance into two components - over-confidence and dismissive behaviour?

Of course we should! Using the same label for two different things is a frequent source of confusion.

I would even add a third component, a "my time is precious" aspect. If there is only one professor of quantum mechanics and thousand youtube-educated quantum magic worshipers, then even if the professor is very humble and if the professor believes these people are nice and good at heart, having a long and detailed discussion with each one of them would be technically impossible, so the professor must optimize for making the debate short, or even avoiding the whole debate.

Okay, in some sense this is also "dismissive", but there is a difference between dismissing people like "go away, you are low status", and dismissing them like "sorry, but there is only one me, and thousands of yous, I can't really give each of you as much time as you would like to get".

There's a lot of nuance to this situation that makes a black-and-white answer difficult, but let's start with the word arrogance. I think the term carries with it a connotation of too much pride; something like when one oversteps the limits of one's domain. For example, the professor saying "You are probably wrong about this" is an entirely different statement (in terms of arrogance) than the enthusiast saying "You are probably wrong about this," because this is a judgement that the professor is well qualified to make. While I can see a person not liking this, I don't think this kind of straight-forwardness is wrong, or arrogant.

The Professor

When I think of an arrogant professor, I don't simply think of a person whose knowledge of a field far superceeds my own. I think of something more than that. I think of a person who seems to have an inflated sense of self-worth because of their knowledge. That is, I think of a person who is overstepping the bounds of their domain; a professor who not only says "You're wrong, and don't have an even basic understanding of quantum mechanics," but does it with an air of regal superiority that implicitly says "Not only are you wrong, but you are not, as a person, worth taking seriously or teaching because you are wrong." That kind of professor is taking the expertise he has in quantum physics and applying it to something well outside quantum physics (in this case, a person's worth). If this is the kind of professor you were describing, then I'd certainly say there's arrogance here. Otherwise, I don't think the professor is being arrogant by saying something like "You don't possess even a basic understanding of the core concepts of quantum physics." Admittedly, a more constructive way to go about this would be to at least show why this is true. As someone who does have an understanding of quantum physics, there's a lot of material that one could easily show an enthusiast that'd be well beyond their current knowledge.

A bit of nuance: Using this same standard for arrogance, the real answer is something like "it depends on what they were talking about." For example, some philosophical theories that emerge from quantum mechanics can be understood and thought about by the laymen (key word: some; many arise from a misunderstanding of quantum mechanics, and thus can easily be debunked with knowledge about quantum mechanics). If the professor and enthusiast were talking about one of these theories, then the professor dismissing this legitimate theory would, in fact, be arrogant (because it would be a step outside of the professor's bounds).

In the case that the professor doesn't admit to being a professor, indeed it would seem arrogant—but only because it seems as if the incognito professor is overstepping his bounds. That is, since no one knows the professor is a professor, there's no reason to assume quantum physics is within his area of expertise. (This may also be somewhat pedantic, but in something like quantum physics, because of this gap in knowledge, it'd be very obvious who the professor was to an audience that doesn't know quantum physics, even if it wasn't made explicitely clear beforehand.)

The Enthusiast

We can apply the same standards of arrogance to the enthusiast. I've seen many people who are simultaneously engaging, calm, polite and arrogant. If a professor, someone whose studied the field very rigorously, with mathematics that would take the enthusiast years to learn, claims that the enthusiast doesn't have a working knowledge of quantum physics—then it's probably true. If the professor says "If you knew enough about quantum physics, you'd see why this idea is strictly wrong," and the enthusiast is still convinced he's correct, then I'd say the enthusiast is stepping outside of his bounds (i.e. - is being arrogant). Again, admittedly, if the professor doesn't explain why the enthusiast is wrong, it's difficult for the enthusiast to gauge whether or not whatever is being spoken of is a legitimate theory or one that dissolves with sufficient knowledge of quantum mechanics.

A bit of nuance: It's easy with quantum mechanics, since so much of it is based in physics and mathematics in ways where the metric for correct or incorrect are fairly straight-forward and clear. If the professor were an economist, the lines defining what the bounds are could be much blurrier.

In the case that the professor doesn't admit to being a professor, the enthusiast might not have a way of knowing whether or not bounds are being overstepped, and the bystanders would probably see the professor as being arrogant. Though, I reiterate: Especially in something like quantum physics, the party who has the expertise would be apparent.

To come back to your distinction between over-confidence and dismissive behavior, I think these things are both addressed by considering arrogance to be synonymous with an overstepping of one's bounds. Dismissive behavior is relevant only insofar as it implies this overstepping. For example, does it indicate that the professor thinks less of the enthusiast as a person? Maybe the professor is being dismissive because the enthusiast's vehemence is blinding? Similarly, over-confidence is relevant only insofar as it implies an overstepping of one's bounds (this is a bit easier, because "over-confidence" is almost synonymous with "stepping over the bounds of your knowledge-limitations.")

Kind-of Disclaimer:

1) I might be underestimating the amount of knowledge you intended for your enthusiast. In my experience (I am not a professor), I've never met a physics-enthusiast who has a working knowledge of the actual physics with which they are enthused—and the types of physics that people find most interesting are usually the ones that require the most knowledge (i.e. - string theory, quantum mechanics, etc.; few non-physicists are that enthused with Newtonian Mechanics!).

2) "Overstepping one's bounds" might be a bad term. I'm willing to use a better one, it's just the one that came to mind. I hope it's clear what I mean from the context.

Positive reinforcement for exactly the sort of post I'd like to see. (specifically, it's starting a useful conversation, and it makes it's point quickly and enjoyably)

I think dividing arrogance into it's component parts is pretty key. I'll hopefully write more on this when I have more time. I also think it's helpful to dive into why (and when) dismissiveness is a problem.

Whoops!