> See how inexact and vague all these measures are. How do you know your confidence is (or should be) 0.5 ( and not 0.49) for example?
Well, you don't, but does it matter? The idea is it is an estimate.
Let me put it this way: we all informally engage in reasoning about how likely it is (given the evidence available to us) that a given proposition is true. The idea is that assigning a numerical estimate to our sense of likelihood can (sometimes) be a helpful tool in carrying out reasoning. I might think "X is slightly more likely than ~X", but do I know whether (for me) p(X) = 0.51 or 0.501 or 0.52? Probably not. But I don't need a precise estimate for an estimate to be helpful. And that's true in many other fields, including things that have nothing to do with probability – "he's about six feet tall" can be useful information even though it isn't accurate to the millimetre.
> Or, how to know you have judged correctly the weight of evidence?
That (largely) doesn't matter from a subjective Bayesian perspective. Epistemic probabilities are just an attempt to numerically estimate the outcome of my own process of weighing the evidence – how "correctly" I've performed that process (per any given standard of correctness) doesn't change the actual result.
From an objective Bayesian perspective, it does – since objective Bayesianism is about, not any individual's actual sense of likelihood, rather what sense of likelihood they ought to have (in that evidential situation), what an idealised perfectly rational agent ought to have (in that evidential situation). But that's arguably a different definition of probability from the subjective Bayesian, so even if you can poke holes in that definition, those holes don't apply to the subjective Bayesian definition.
> Or how do you know the transition from "knowledge about this event" to "what it indicates about its probability" you make in your mind is valid?
I feel like you are mixing up subjective Bayesianism and objective Bayesianism and failing to carefully distinguish them in your argument.
> But in any case, the significance of new evidence still has to be interpreted; there is no objective interpretation, is there?.
Well, objective Bayesianism requires there be some objective standard of rationality, subjective Bayesianism doesn't (or, to the extent that it does, the kind of objective rationality it requires is a lot weaker, mere avoidance of blatant inconsistency, and the minimal degree of rationality needed to coherently engage in discourse and mathematics.)