In general you are correct that stay away and keep away can be used interchangeably without loss of understanding, meaning don't go near. Stay away seems more AmE while keep away seems more BrE.
One way of thinking about the distances is to see usage of the keywords in other related phrases. As an native AmE speaker, my thinking behind my sense of the differences in distances involved is
far from is the most distance, as it is indeterminate
far from us, another civilisation may exist
keep far away is further away than keep away
keep away is the closest, keep also has the meaning of possession which can imply things being at-hand or within reach
keep clear of the moving train
keep to yourself
stay away is somewhere in-between and possibly inclusive of far away's distance
He pointed and said "stay over there".
One would never say keep over there, thereby giving stay a farther meaning.
Stay far away and keep far away are interchangeable for me since the distances inherit the magnitude of far
For stay away, keep away, and far away the quantities involved are not as easily defined as the other descriptive adjectives a few, some, or several