Counting the Number of Closed Binary Operations That Are Commutative

You are right. Here is my way to see it with the following arrays that are clearly in a bijective correspondence with the binary operations.

A $5 \times 5$ array of a commutative operation has this form:

$$\begin{array}{c|ccccc} &a&b&c&d&e\\ \hline a&s_1&r_1&r_2&r_3&r_4\\ b&r_1&s_2&r_5&r_6&r_7\\ c&r_2&r_5&s_3&r_8&r_{9}\\ d&r_3&r_6&r_8&s_4&r_{10}\\ e&r_4&r_7&r_9&r_{10}&s_5\\ \end{array} $$

where we are free to place, for each of the 15 "sites" $r_1,\cdots r_{10}$ and $s_1,\cdots s_5$, one of the five elements $a,b,c,d,e$, giving rise to your solution $5^{15}$.

The solution given by the book looks to forget the diagonal choices.

Edit: I just saw a nice general answer here.