If you are referring to a measure then 'almost all' pretty much exclusively means every except for a zero-measure set. I've never encountered another definition when dealing with measures.
But consider a different, entirely valid context of "almost all":
"Almost all natural numbers are greater than 10."
"Almost all prime numbers are odd."
If we wanted to extend this intuitively, we might want to support the statement that "almost all positive reals are greater than 10". One option of doing that is by using the nonstandard definition of "everywhere but on a finite-measure set".
But consider a different, entirely valid context of "almost all":
"Almost all natural numbers are greater than 10."
"Almost all prime numbers are odd."
If we wanted to extend this intuitively, we might want to support the statement that "almost all positive reals are greater than 10". One option of doing that is by using the nonstandard definition of "everywhere but on a finite-measure set".