As to Chair and Chairman and Chairwoman and Chairperson (and maybe others), according to a ruling by the American Parliamentary Association (most of whose members are female, by the way) the proper designation for the person who runs a meeting is "Chairman", with the option of "Chairwoman" if the gender of the occupant is of that persuasion and she desires the distinction. Yes, "Chairperson" is possible to refer to the incumbent, I suppose, but it's awkward and unnecessary.

Why?

Because "the Chair" of an organization or meeting is not a person but an office or an organizational function. The person who exercises the function is properly called "the Chairman" or "Chairwoman". "Man" here, as in many other "-man" words, does not assert a gender at all, but merely that a human being does it. That's the meaning of "man" in such words, and has been for eons.

To say, "She is the Chair of the committee/association/company/etc." is simply wrong.

Dave Oesterreich