Constructing inequivalent binary matrices
I have made some progress towards answering this question. I am posting here in case anyone else is interested and also because this construction might have some usefulness for (directed) graphs.

Count the number of 1s in each row. Let $a_0$ be the number of rows with zero 1s, $a_1$ the number of rows with one 1 and so on up to $a_8$ which is the number of rows that have eight 1s. Obviously $\sum a_i=8$. The proposed parametrization that I have come to after trial and error is:
$$(a_1,\cdots, a_8; T, S)$$
where T is the trace of the matrix and S is $1$ if the matrix is symmetric and $0$ otherwise.
T runs from $0$ to $\sum_{i=1}^8 a_i=8-a_0$.

From my trials and errors it looks like that if two matrices are different in this parametrization then they belong to different equivalence classes, so to construct a representative in each class we just scan through the space of parameters as described above. 

(Update) It turns out that this parametrization works fine for n=2 but not for n=3 as it can be seen by a brute force calculation. I still think it provides some insight on the structure of the answer and I invite people to try and modify/extend it to cover the most general case.