They have the same standard of proof but different onus
The legal system places the onus of proving an allegation on the person making the allegation. For your example, this is A if they are suing B or the government prosecutor if B is being prosecuted.
The standard of proof is “beyond reasonable doubt” if B is defending a criminal prosecution and “balance of probabilities” if B is defending a civil prosecution or a lawsuit. Courts have historically been reluctant to define these terms further because doing so can lay grounds for an appeal if the judge oversteps so they mean what their plain English formulation means and what they mean precisely in any given case is one of the things the trier of fact has to decide. For A or the prosecutor to win, they have to meet this burden for each and every element of the offence or cause of action; if they don’t, then B wins.
So, B doesn’t have to offer a defence at all and will still win if A doesn’t meet their burden. If B does offer a defence then the trier of fact compares the evidence of each side and decides which they prefer and therefore whether A has met their burden. A jury doesn’t have to give reasons for their decision; a judge does. In general, the decisions of the trier of fact are not appealable unless there was no reasonable basis in evidence to support the decision. For example, the evidence of B might not be believed - this is fine, unless the reason for not believing it is that B has a beard and everyone with beards are liars.
Options 2 and 3 are simple matters of comparing evidence and deciding which is preferred. Option 1 is different; it is what’s called an affirmative defence. Here, A has met their burden because B conceded. Now B is relying on the position that they had a legal excuse, that is B is alleging a position and B has the burden of proving it. Now, the burden on B is always “balance of probabilities” because B is not alleging that anyone committed a crime so they don’t have to reach the criminal standard.