To outstay vs. overstay one's welcome

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?

According to the American English corpus it seems the idiom began as outstay one's welcome but in the last thirty years the version with overstay (red line) is by far the most common.

A different story is told by the British English corpus where it appears that in the early 20th century outstay (blue line) was clearly the favourite until the 1980s, when (I imagine) American books, magazines etc. were increasingly published and printed in the UK. Of course I have no hard evidence to prove this supposition but it explains why my English 57-year-old companion is convinced that outstay is the "correct" version. Personally, I prefer overstay for the reasons I pointed out in my original question.

A welcome, a real welcome, is something a host gives their guest. The nature of the welcome - what it includes and how long it lasts - is entirely for the host to decide.

Overstay has the sense of going beyond a known limit. A guest cannot overstay their welcome since they do not know where the limits of their welcome are ? The most a guest can do, while they are a guest, is hope that they do not outstay their welcome (ie stay longer than they are wanted, however long that may be).

Outstaying is a highly nuanced word that will vary hugely on circumstance and personnel (some guests outstay their welcome sooner than others!). Overstaying is a much more prosaic word. It can be used only if the limits of a welcome are known.

I think the difference here is likely due to the different metaphors being used to represent a person's "welcome."

We say you outstay your welcome to evoke the image of one's "welcome" being a presence/atmosphere staying there in the place with you, and then fading or leaving once the welcome (the sense of hospitality from your hosts) is no longer present. In this case, if you are still staying, you have literally outstayed your welcome in the comparative sense of "out-verbing" someone, because your welcome has left and you have stayed longer.

We say you overstay your welcome to evoke the image of one's welcome as a "capacity" being provided by your hosts which you are filling up by your stay. When you have reached the limit of that capacity, you are figuratively "overstaying" the welcome "bucket" which your hosts provided.

I generally think of the latter formulation as implying a somewhat more fixed quantity of availability while the former implies more of a variable or open-ended length of welcome that depends more on the mood of your hosts than any prearranged length. When I watch British period dramas, it seems to me that "summering with remote relatives for a variable amount of time" was a fairly well-known tradition there, while in America we mostly stay in hotels, with prearranged lengths. Even when summering with relatives, we tend to arrange fixed lengths of stay in advance. This may account for some of the difference in usage between the countries.