Correct But Too Loose

Neither. The right statement is $\Theta(n)$ for the indexing plus $\Theta(\log n)$ for the comparisons.

Both statements are true. $O$ is an upper bound and $\log n \le n$, so $O(n)$ holds; the reviewer's is the tighter one and the one worth writing down.

The reviewer. An algorithm has exactly one big-O class, so naming a different one is an error.

The author. $O(n)$ is right because binary search can inspect up to $n$ elements when the array has duplicates.