I know what you mean, and I say exactly the same as you. This Friday is - well - this coming, Friday, and next Friday is the one after that. OH thinks the same as your client - Next is The Next One.
OH also has a habit of counting the day from midnight, so if we get into bed 00.05 and I ask him what he's doing tomorrow, i.e. the coming day, he'll tell me what he's doing the day after that, on the basis that the tomorrow I'm talking about is actually today.....(have you lost the will to live yet?)