I came across the expression "outstayed my welcome" in the following excerpt of a novel
I glance around and see that the café has filled up with people ordering lunch and that a couple is queuing by the door. I have outstayed my welcome.
p53, Apple Tree Yard By Louise Doughty
I have no difficulty in understanding the idiom, which means to stay in a place longer than one should or is invited to; Oxford Dictionaries also includes the verb overstay and gives the following examples
he makes you feel you’ve outstayed your welcome before you’ve even sat down.
Finally the moment came when I knew I had to leave as I had already stayed for dinner and overstayed my welcome.
However, I understand the term "overstay" much better, it makes more sense to me because the prefix over- is used to express an excess of something or the idea of "too much" e.g. overweight, overdone (when a piece of meat has been cooked too long), oversleep and overtime, but I have difficulty with the prefix out-. We don't say outweight, outdone (referring to food), outsleep nor outtime, so why do we say "outstay"? To me the latter seems to express an endurance test, as if I stayed in a place longer than anyone else. Am I mistaken?
Is "outstay one's welcome" more common than "overstay"? Is there any difference in meaning or are they completely interchangeable?