A claim is just a belief stated out loud.
I would disagree.
Do you believe that the multiverse theory is true?
Depending on what you answer, I would be in the right to say that you are making a claim for that belief. And therefore also that you have a burden of proof.
Do you think Im being reasonable?
It's not necessarily either. "I don't believe gods exist" can be an expression of what you don't believe. Depending on the context, it could also be a way of expressing a claim that no gods exist in a way that's consistent with social norms.
To avoid this confusion, I'll sometimes express this idea as "the number of gods I believe in is zero."
No, because a belief only deals with the assumption of what one think/believe to be true. Doesn't say anything about whether that belief is true or not.
It comes in various degrees of certainty depending on the amount of evidence supporting the belief. Which can range from nothing to a lot.
An example could be intelligent life in the Universe, it is not unreasonable to believe that it could be possible. The Universe is huge, there are lots of planets and we have evidence that it at least can appear using ourselves as evidence. Yet, despite this, it is far from certain that other intelligent life exists out there. So even if you believe it to be likely, it is not a claim that other intelligent life exists, it is merely what you find to be the most plausible based on the limited information you have.
The more evidence you can add to support a belief the more rational it is, even if it turns out to be wrong in the end.
If I'm getting into an in-depth discussion, I'll sometimes note that there are several different ways to not accept belief in a deity (or anything, really):
1. Not be aware of it. I haven't even heard of most of humanity's gods, so of course I haven't accepted that any of these gods exist. No belief involved yet.
2. Not accept the argument without rejecting it: responding to the argument for God with something like "huh?" or "I don't follow" or "sorry - I don't speak your language". Still no belief involved.
3. Reject the argument: respond to the argument with something like "your reasoning is wrong" or "that's nonsense" or "you haven't met your burden of proof." There's now a meta-belief about the argument for the god, but belief about the god still isn't in the picture.
4. Reject the conclusion: "I know that your argument is wrong because we know (or there's good evidence that) your god doesn't exist." Belief in the existence or non-existence of the god is now involved.
How you choose to answer or approach it, I think is irrelevant. When talking about beliefs there are 2(3) options.
1. You do believe / agree
2. You don't believe. / don't agree
(3. You don't know. (Could make an argument that this is redundant.)
It doesn't matter which belief I present you with, it could be:
"I believe there is life on Pluto, do you agree?"
Ultimately it doesn't matter how you reject or approach my belief. In the end, it will be one of those two options, you either agree or you don't. The important thing here, is that you are not evaluating a claim, simply whether you agree with my belief or not.
It is first after this step that we get into the claims.
"So you might ask me, why I believe that?" To which I will start adding things that I think support my belief. These are the claims that require a burden of proof.
"Baha'u'llah was a messenger of God" relies on the existence of God. If God doesn't exist, the claim must be false.
Yes, It depends on how they initialize the argument.
So they could start by saying that "I believe in God" or "I believe Baha'u'llah was a messenger of God", and as you say, it will lead to them having to justify why that is the case, which means that they will have to present a claim that makes their belief rational if they want anyone to take them seriously.