We're saying approximately the same thing, though I'd argue that your phrasing is slightly more confusing to people that aren't super familiar with Big O notation (and therefore don't exactly understand what is meant by "ignores constant factors" and similar).
Reads from the original two matrices and writes to the final matrix aren't at all the crux of the problem, so I don't think it makes sense to focus on either of those things in your explanation. The goal of the problem is to reduce the number of multiplications which are required to find the values that you then write (of which there will always be n^2 writes). There will also be 2n^2 reads, but likewise that isn't relevant because it's not what you're trying to optimize.
The relevant idea here is that the naive method of multiplying a matrix involves n multiplications for every row-column pair, for a total time complexity of n^3 (n rows, n cols, n multiplications). The goal here is to bring down that third number, from n down to something >= 1, with the holy grail obviously being 1 itself.