The CFAI text mentions compensating directors appropriately, but doesn’t talk about this specific issue: are directors who own the shares of the firm on whose board they serve considered independent?
Example: Eric Schmidt sitting on Apple’s board, getting compensated in Apple’s stock. Is Eric independent?
As far as I know, “Eric” would not be independent. However, this does not mean he can’t sit on the board. You need to look at the board as a whole to address independence. SEC rules require the majority of the board to be independent. Also several of the committees that are formed by the committee have to be independent. Check SEC rules, Specifically DF Act.