What does actually probability mean?

You dip your toe into the waters of controversy with such a question: the quoted definition is the frequentist definition of probability, the second one is called Bayesian, and an excellent way to cause a brawl among statisticians and probabilists is to ask them which they are: see Wikipedia's article for a long summary. There's plenty of other articles around the Internet that talk about this as well, but I thought that this question on Statistics StackExchange was a good explanation as well.

To put this in a physicist's context, it's about as controversial as the interpretation of quantum mechanics (except no statistician is crazy enough to suggest a many-worlds equivalent).

There are actually two competing notions of probability, Frequentist and Bayesian. Frequentist probability refers to the notion involving the frequency of a result in repeated trials; Bayesian probability is roughly a measure of our belief or confidence in an outcome occurring. Bayesian probability used to dominate, but Frequentist statistics is much more common today.

In the case of quantum mechanics, the Frequentist interpretation is considered the more natural one by most physicists: quantum mechanics is inherently non-deterministic, and the exact same experiment can produce different outcomes. Quantum mechanics only allows you to calculate the frequency of the various possible outcomes.