In the preceding chapter, we have calculated the chances of an event, knowing the circumstances 
under which it is to happen or fail. We are now to place ourselves in an inverted position: we know the event,
and ask what is the probability which results from the event in favour of of any set of circumstances under 
which the same might have happened.

De Morgan (1838), An Essay on Probabilities, Ch. 3 "On Inverse Probabilities"

What was not clear about this term in the past?

The inventors of the term seemed to not notice the subtle difference between the probability of a real event and the "probability of the probability of an event having some particular value". The tautologically equivalent word 'likelihood' (instead of "probability of the probabitiy" of course) doesn't clarify the situation, it makes it even more vague and obscure.

What do we do about this confusion?

Eigther we continue using this word 'Bayesian' all the time until the end of days or... Let's introduce an operator with a simple and obvious name 'inversion'. If you apply this operator to a probability distribution function of some data you get an inverse probability distribution function. In the same way as you calculate an inverse function from a functional equation y = f(x) by applying the functional inversion operator (.)^-1 and getting an inverse function f^-1(y) and equate it to x because it is 'obvious' (or 'defined' somewhere) that f^-1(f(x)) = x.