I don't understood how he says that he's not male, but then insists on this distinction between cis women and trans women. ONLY males can be trans women. If he wasn't male, he wouldn't be a trans woman - he would be a cis woman (by his lights). He can't have it both ways - he can't be a female trans woman.
Honestly, I would just say to him, right lets use the terms cis woman and trans woman, but let's be absolutely clear about what these mean and get him to give a definition.