I think John has more or less discussed the matter appropriately. S^2 is
what one wants to calculate most of the time. I usually just simply
don't worry about its name, which means my relativity class catches me
being inconsistent from time-to-time with the word "interval". Here is
a case where the nomenclature, even inconsistently used, rarely causes
problems.
Mathematically they are equivalent, of course; so this is mostly a
matter of taste. (You can often tell what the academic background of
one's first relativity instructor, by which metric the student uses in
later life.)
My preference naturally comes from how I learned it the first time :-)
Combined with a the idea that the positive signature metric matches the
formula for distance squared between two points in 3dim Euclidean space;
i.e. the displacement between the points "dotted" with itself; when
(DELTA t) equals zero.