If we believe in the standard conjectures (or something similar) so that the category of motives is Tannakian, then a Weil cohomology theory is just a fibre functor and as such twists of each other (that does not quite give the multiplicative structure however) which gives, more or less, a classification. Note that your question on Grothendieck sites is only vaguely related, the whole point of the notion of Weil cohomology theory is that there is no site in sight (pun intended).
Addendum: This is a little bit simple minded as the requirements for a Weil cohomology theory to be a fibre functor go beyond the standard conjectures I think. Also one would need to define motives using cohomological equivalence. However, I think the statements I made are philosophically OK and one cannot hope to get anything in the way of a more precise classification.