That is exactly what I mean. We try to establish what a probability of 0.5 means, i.e. that the relative frequency converges to 0.5 if the number of trails goes

That is exactly what I mean. We try to establish what a probability of 0.5 means, i.e. that the relative frequency converges to 0.5 if the number of trails goes to infinity, but that is not exactly true because there is a (possibly vanishingly small) set of outcomes where the relative frequency does not converge to 0.5. This in turn forces us to state that the relative frequency only converges with high probability, but now we have used a probability in our definition of probability, i.e. we made some kind of circular argument. I just skimmed the article you linked and I am not sure if I read it before so I will read it again, but from a first glance it does not look like any definition in there is suitable to avoid the problem.