the word “hermaphrodite “is considered a slur in the intersex world, it’s quite old-fashioned and frowned upon.
but yeah, intersex folks are not a third sex. They are dimorphism gone wrong. Or gone off a little in most cases. There are all kinds of curious anomalies in sex markers but they are anomalies.
Common thought among intersex people is that there are many more intersex people out there in the population than is commonly believed. Why, there are as many intersex people as there are redheads! Or so we are expected to think.
But in the end, the vast majority of those intersex people can line up rather easily with male or female people.
You are right, there’s not a third sex and intersex people are not reproducing to make more intersex people just like them.