I would like to be sure about the correct interpretation of the p-value considering the $\chi^2$ test for a fit.
I know that the null-hypotesis must be accept or refuse based on whether the p-value is greater or less than the significance level. However it is correct to say that it is the probability that if the experiment were performed again then you would get a bigger $\chi^2$ and then a fit tha less corresponds the data. I would like to be sure about the practical interpretation of this parameter.
You are on the right track.
You only need to add, that the $p$-value is the probability of getting a larger $\chi^2$ than $\chi_{sample}^2$ under the assumption that $H_0$ is true:
$$p = P(\chi^2 > \chi_{sample}^2 \, | H_0)$$