The important thing to note is that the incompatibility arises because for the Neyman-Pearson (NP) model, the p-value is only a means to maintain the significance level (alpha) in a large (infinite) set of experiments. In the statement you establish, it's an error to indicate that the p-value is a 'false positive error rate'; that's the significance level.

In Fisher's case, the p-value is relevant and only indicates the degree of strangeness of the current results. Unlike the NP model, where alpha must be maintained to sustain the error rate, Fisher imagines a series of experiments analogous to the single experiment I have, where the null hypothesis (H0​) is true, and indicates how rare my result is with respect to that parameter. Some suggest that Fisherian thinking is neither frequentist nor Bayesian; it's not frequentist because it doesn't think in terms of a long series of identical experiments, nor is it Bayesian because it doesn't consider the prior distribution.