I guess we are just going to have to agree to disagree. Because I have also worked in hospitals. Not as long as you, but I do have that experience. Maybe it is a cultural difference. Or possibly a state/city difference, (not knowing where you lived).I too am from the west coast and worked in hospitals most of my adult life. A male nurse is a murse. A satchel is a man purse, but no-one actually says satchel, it's called a "beg."
I will concede that satchel is the correct term for it. However, the only people who actually call it that are the ones who wear them and their friends. Including using just the word bag.