The advice I have received is that there is no definitive answer to this question. It is all a matter of interpretation, since there is more than one piece of legislation and the words used are not so explicit as to remove doubt. There has not been a relevant court case to provide a precedent.
However, the view that a co-opted member can be chairman (or vice chairman) seems to have the greater merit in terms of reasonableness and practicality, especially where a council has no elected members.
It is clear, though, that a co-opted member cannot claim an allowance, including a chairman's allowance. The treatment of allowances is the only area where there is definitely a distinction between a co-opted councillor and one who has put up for election.