When you say you are sick then you are sick, which could mean physical or mental. Yes it is usually easier to talk about physical illness than mental but some physical illnesses might be just as uncomfortable to disclose. For example: “I can’t work because I am in a lot of pain because my hemoroid flared up bad and i can’t sit , my chronic STD flared up and it hurts badly (or itches) etc etc” You’d likely lie in those cases too.

In general unless they insist on you providing details, you don’t need to disclose neither physical nor mental discomfort