I've stumbled upon the book "Introduction to probability" by Anderson, Seppalainen, and Valko. It's a college textbook.
Right in the second chapter it does a very good introduction into the Bayesian formula, deriving it from the descriptions of weights of events, in the same way as I struggled to reinvent on my own. And right there it also provides the "General version of Bayes' formula" for the enumerated events, that took me so much time to reinvent on my own. Well, another case of "everything is invented before us" :-) If only we knew, what books to read at the right time :-) On the other hand, I'm not sure if I would have recognized the importance of that formula if I didn't have a use for it in mind first.
In case if you wonder, here is the formula:
If B1...Bn are events that partition the sample space, then for any event A we have:
P(A) = sum from i=1 to n of (P(A & Bi)) = sum from i=1 to n of (P(A|Bi)*P(Bi))
P(Bk|A) = P(A & Bk) / P(A) = P(A|Bk)*P(Bk) / (sum from i=1 to n of (P(A|Bi)*P(Bi)))