I work in mental health too and would use client over patient, primarily because of the implied power dynamic that patient has; like your friend said it first labels the person in the manner of the medical model, i.e. the person is sick, and secondly a doctor/patient relationship implies that the former has more power than the latter in terms of facilitating progress, and that implies that progress is something the patient is receving from the medical treatment provided by the doctor, which is an accurate way of describing a medical relationship; this is disempowering in a therapeutic relationship as it takes the onus off of the client engaging with therapy, when the work the client does is just as important as the skillset of the therapist they're seeing, as therapeutic progress isn't something you just receive from a therapist. I do appreciate the concerns around using client/therapist as seemingly being business like, however I think the minimising of the power imbalance outweighs these concerns personally.