Is there an error in the one-sided binomial test in R?

I am not sure, but I think R is doing not what it is supposed to do. Using binom.test() my understanding of the parameter alternative="greater" is the following hypothesis. $$H_0: p \le p_0 \quad \text{vs.} \quad H_1: p > p_0$$ (I mean it says greater and not greater or equal).

P-Value then should be calculated as follows: $$p(c) = P_{H_0}(Z>c) = 1 - P_{H_0}(Z\le c)$$

Here is the equivalent R-Code

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