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As a concrete example, consider Missouri v. Coleman, where Coleman handed a teller a plastic bag and said "I need you to do me a favor. Put the money in this bag", and later as the manager approached, said "Ma'am, stop where you are and don't move any farther". Coleman was convicted of second degree robbery: but the appeals court found that he had not acted forcibly, so his conviction was overturned. Instead, the court entered a conviction for the lesser offense of stealing which is when one

Appropriates property or services of another with the purpose to
deprive him or her thereof, either without his or her consent or by
means of deceit or coercion

The act may also be termed "theft", as in Washington state. The thief is acting deceptively and thereby gaining control over property.

Generally, the law would not just look at the robber's literal words, but at how a reasonable person would understand them in context. And it will assume that the robber meant them to be understood in that way. Here, a reasonable person would understand such a note to be a threat of violence, so the law will assume the robber meant it as a threat.

Likewise, a mob boss who tells his associates to "take Joe for a ride" will not be able to avoid prosecution by insisting that he only told them to provide him with a pleasant sightseeing tour.

How is this different than say people asking for charity donations in
the street?

Context is everything.

Obviously, someone asking for charity donations on the street will present themselves as a legal charity, like the Salvation Army, etc. If you meant "charity" in the sense of someone panhandling Panhandling | (Merriam-Webster), most if not all people will take the asking for $1000 as an absurd request and keep walking. It's possible that someone may take offense and the police may deem it an attempt at mugging or stealing, depending on jurisdiction, and see user6726's answer.

But to walk into a bank and ask for $1000, the context is completely different. That's because a bank is where the money is. You don't usually walk into a bank for anything other than a financial transaction, either a legal or an attempted illegal transaction. As the (apocryphal) saying goes, Willie Sutton (Wikipedia) robbed banks "Because that's where the money is".

So while it's possible that a polite bank robber trying to not be aggressive may get away with robbery under a lesser charge, anyone in a bank requesting an illegal withdrawal - by any means - is still a bank robber, and 99% of the time, I think a jury and judge will see that.

You can't have it both ways. Either the person used words or actions that, under the circumstances, caused the teller to reasonably fear that violence will occur if the money isn't given or they didn't. If the former, it's a threat. If the latter, the teller won't give them the money.

If their words caused the teller to give them money because the teller feared violence, and this was the intended result of their words and not totally unexpected, that's the definition of using a threat of force to obtain a thing of value.

The lack of use of force means, of course, that it's not a violent robbery. But if a reasonable teller feels threatened under the circumstances, it's a threat. If the teller doesn't, they won't give the person the money.

Robbery is theft carried out through force or the threat of force. The most important part of your hypothetical is that you're saying the person asks for money without making a threat (and presumably without force).

If that is the case, the person is not a robber and the transaction is not a robbery. It is, like you said, the equivalent of a request for a charitable donation, and it is legal.

IMHO one factor which would be considered in some jurisdictions when deciding if a crime was committed in your scenario is that the cash in a cashier's drawer and in the bank vault is either the property of various depositors or else the property of the corporation that owns the bank, and thus of its shareholders, or possibly a mixture of the property of both.

The money acquired in your scenario is not the property of the cashier or of the bank manager. Thus the cashier and the bank manager don't have unlimited authority to hand out money like they would if it was their own money. Instead they are restricted to handing out money to those that it is their bank's policy to hand out money to, and in the amounts that it is their bank's policy.

Thus a person who makes bank employees pay him money he is not entitled to is making them steal money from its rightful owners. He is making them commit a crime. If person A tries to make person B commit a crime that person B wouldn't want to, and if person A succeeds in making person B commit that crime, person A should be considered to be equally guilty of the crime he makes person B commit, and he would be considered equally guilty inmost or all legal jurisdictions.

But what if the bank has a policy of giving the bank robber what he asks for and not making any trouble for him in hope that he leaves without harming any one? In that case the bank employees are following bank policy and doing as authorized. Therefore they are not stealing from the bank and the robber hasn't forced them to steal from the bank but has made a withdrawal that doesn't violate bank policy. So the bank robber shouldn't have committed any crimes, right?

Except that the nonviolent bank robber has let the bank employees think that he might be, and probably is, an evil, violent, homicidal bank robber. And thus he has acquired money from the bank by lying by omission about being a nonviolent bank robber, and thus by fraud, which is stealing. So the nonviolent bank robber is a criminal of a different type and might be given too options when arrested. He might testify that he is nonviolent and would never have hurt anyone, and thus be convicted of stealing by fraud, or else testify that he is violent and would have used force if opposed, and thus be convicted of using the threat of violence to commit bank robbery.

I suspect that any legal loophole that anyone can think of has been thought of before by criminals and lawmakers and has been closed by legistation and/or legal precedents in various trials.