A 'critical thinker' is not looking to achieve a "belief", at all. A critical thinker is looking to dispel a belief.
I don't know what you mean by belief. Some people distinguish it from knowledge. I don't. For me, a belief is anything I consider to be true. Having defined what I mean by belief, which may not be what you mean, then I have to disagree with your comment. Critical thinking is how the critical thinker comes by his beliefs. He either applies reason to evidence to arrive at sound conclusions, which become new beliefs that are considered correct, or he reviews the arguments of others for soundness, and if he finds them compelling, acquires a new belief and increases his fund of knowledge.
Basically, you're saying that "if they can't convince me of it, it isn't so"
No. I am saying that if I can't be convinced of it, I don't believe it. You seem to be incapable of making this distinction between I don't believe it and I claim that it is false. You've converted the first, which is my position, to the second, which is not, and I don't think you realize that the two are different. I say that because there is never anything in your writing to indicate that you can conceive of these ideas as distinct. You flip from one to the other as if they are synonymous and interchangeable.
If you had ever said something like, "I don't believe that you merely don't believe that gods exist as you claim. I believe that you actually believe that gods don't exist." A comment like that would indicate that you understand the distinction between what the agnostic atheist claims and what the typical theist converts it to. But instead, you just slip from one to the other without any indication that you realize you did or that the two are different.
Actually, it does. At least it includes every atheist that claims that they can't accept the existence of a God that they can't know to exist by objective, evidential proof. That's a very 'gnostic' position if ever there was one. And pretty much every atheist I've ever encountered makes this claim.
And there you go again conflating agnosticism for gods with gnosticism. These are distinct ideas to some people. The gnostic theist says there is a god. The gnostic atheist says none exist. The agnostic says neither of these things. You described agnosticism for gods in the skeptic, then called it the opposite.
Are you aware that you do that? I think not, but if you are, why do you do it?
None of this requires anyone to prove or "demonstrate" anything, because the assertions are all subject to the individual making them. They are SUBJECTIVE assertions. "Peas are fantastic!" "God is love!" "Evolution is fiction!" Whatever. We can discuss why we make statements like this, but no one is obliged to prove their veracity to anyone else because they're just subjective opinions.
I've already said that opinions don't need to be supported unless one considers them demonstrably true, wants to be believed, and is dealing with a prepared mind capable of critically evaluating an argument and open to accepting a compelling one. I think that most critical thinkers understand that others' opinions about the taste of peas aren't arguable. Regarding evolution being a fiction, that is an existential claim that can be refuted unless it is not stated as an unsupported belief using the words, "I believe" or "In my opinion." Do that, and my answer is, "Well, that's not my belief or opinion" or no response at all, just as with him liking peas.
It remains the case that if you wish to convince a critical thinker of anything he doesn't already believe, you'll need to make a compelling argument, which is what saying he has a burden of proof actually means. If you don't do that, he treats the claim or unsound conclusion as an opinion which is not accepted as true (believed, as defined above).
It doesn't matter if they want to be believed or not. All that matters is if they are proclaiming a universal truth: a truth that we all are expected to accept as such. Then, they are obliged to offer proof that can stand up to critical review.
So they utter what they consider to be a universal truth, one which they expect others to accept as such, but at the same time it doesn't matter if they want to be believed or not? That is doomed to fail if dealing with a critical thinker who requires justification before believing. If one doesn't care if he's believed, he won't try to convince, and he won't convince. If he expects the critical thinker to accept this claim of a universal truth with which he doesn't already agree, he'll need that compelling argument.
that doesn't mean that you must be convinced for it to be considered "proof".
Disagree. "Proof," by which I mean not proof as a mathematician means it, but a convincing demonstration, is that which convinces. It's as meaningless for somebody to call his argument that convinces nobody proof as it is for a comedian who makes nobody laugh to call his act funny.