Yet the first 3 films are a trilogy, telling the story of Ash. Once they added the remake, the series became a franchise.
Surely the idea of a continued, self-contained story is more important than the number of movies? Teproc is right that, although the Indiana Jones movies, of which there are only three, make up a trilogy, that trilogy does not have an intermovie arc. They are three self-contained movies that take place in the same universe and feature the same characters. No plot points otherwise connect them. By your own rules, it should count as a franchise.
Remakes on the other hand, shouldn't. They reset the continuity and therefore belong to, if anything, a new franchise. The Batman movies, for example, do not share continuity or universe, so should not be considered a franchise, including the Nolan ones, which tell a unified story, and are therefore simply a trilogy.