If you see no difference between saying "it either rains or it doesn't" and saying that some days it's more likely to rain than other days I wonder why would you ever decide take an umbrella with you in case it rains later.

But that's fine! This line of discussion seems exhausted.

You are missing the point completely.

You don't make a probabilistic calculation based on some formula to figure out whether you bring your umbrella with you or not.

You just guess, sometime based on past experience.

Sometime you are right, sometimes you are wrong. But you are always just guessing and no formula will allow you to end up with whatever probability you calculate.

Please correct me if I'm wrong. I may have misunderstood you. You say that:

A) there is no difference between saying "It either happens or it doesn't" and "There is a 40% chance it happens"

and, I assume, you also say that:

B) there is no difference between saying "It either happens or it doesn't" and "There is a 60% chance it happens"

So - unless you also have your own ideas of how logic works - you will also say that:

C) there is no difference between saying "There is a 40% chance it happens" and "There is a 60% chance it happens"

If there is no difference how could it be based on past experience or whatever?

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But maybe you didn't mean that and you recognize that saying that the probability is 60% means saying that it's more probable than when it's said that the probability is 40%.

It's clear that you don't like people using the word probability. However, the fact is that this use is completely in line with the mathematical definition of probability and with the meaning and etymology of "probable" (likely, reasonable, plausible, having more evidence for than against).

"C) there is no difference between saying "There is a 40% chance it happens" and "There is a 60% chance it happens"

AND

"If there is no difference how could it be based on past experience or whatever?"

These are two different propositions and questions.

Past experiences can tell me that if I see dark clouds then it could mean rain.

I could be wrong, I could be right, but I have no way of calculating the probability of that.

You can just guess, sometimes based on past experience! The same thing that you do already - but calling that probability as many other people do. There is no shame in saying things like “it’s twice as likely to happen as not”, assigning the numerical values 2/3 and 1/3 and using probability theory to work on those numbers.