I definitely agree that taking a sick day is justified - a psychiatrist is a doctor and mental illness is still a health issue that impacts on work. I also agree that your boss and colleagues doesn't need all the details. At most they can ask for a medical certificate to prove that you saw a medical professional and that would only give the doctors name, the date off and a comment about why you were off work (eg. 'unable to work on [date] due to medical assessment / treatment / illness').

I hope it all goes well!