Null hypotheses are phrased in terms of presumed stochastic data-generating mechanism; they do not address the data directly. That said, you are right about the conclusion one is to draw from the test. Fisher himself phrased it as
Either the hypothesis is untrue, or the value of [the test statistic] has attained by chance an exceptionally high value. [emphasis in original as quoted here].
Null hypotheses are phrased in terms of presumed stochastic data-generating mechanism; they do not address the data directly. That said, you are right about the conclusion one is to draw from the test. Fisher himself phrased it as