(1).a quorum of at least half of the 20 directors must pass the resolution; (2).by at least a two-thirds majority;
the former is the main part and tells us the main idea that a quorum must pass the resolution. we know that the quorum must be at least half of the 20 directors. the verb is not " present" but "pass", so, the "a quorum" passes the resolution, but, not present the vote meeting;
the latter is telling us a way in which the resolution will be passed, and this way is a necessity, a requirement must be met. at least a 2/3 majority. meaning those directors who pass the resolution must be more than 2/3 of the totle directors who pass, refuse or give up the resolution.
the above is my personal cognition.