What do you think the purpose of the "by or on behalf of the employer" phrasing is intended for if not that
I think that could include where payment is made, for example, by an agent on behalf of the employer. I do not think it is wide enough to cover a situation where a third party agrees to pay the cost to the employer, as that is caught by the concluding words of 130(9)(a): "provided that neither the pilot in command nor any other person who is carried
is legally obliged, whether under a contract or otherwise, to be carried.
So, if A (client or customer) pays C (employer) so that B (pilot) makes the flight that is valuable consideration and caught both by 130(1)(a) and 130(9), as there would be an agreement, i.e. a contract, in place. Whether the pilot is re-imbursed any costs of the flight he has paid by C is irrelevent in this scenario as it is the valuable consideration paid by A which takes it out of the private flight category.