The confusion here is based on the difference in understanding between a formal position (chair) and the relative power of a boss.

There is no official chair of the commission. But each families (and hence boss) power is also different. Resulting in boss's differing powers of influence. That is to say though Chin, for example, never officially held a formal 'chair' role, his power and hence influence was such that he could be considered to have held the defacto role.