Since I work in the entertainment industry (mostly in theatre), I think it's actually valuable to look at the etymology of why we call them "scripts" in the first place. I always made the connection that a script is a series of commands that tell a bunch of programs how to work together to make something meaningful happen.

Have you seen live improv theatre? Sometimes, and only very rarely and/or with very good actors, does it ever work. Actors tend not to do very well on their own. That's why we have playwrights and directors to help them out. You can toss a bunch of them up on stage and let them play, just like you can have a bunch of executables in your path, but they all work so much better when you have something telling them what to do.

So a script relies on external programs to do the bulk of its work, and a program does not. We have a lot of modules and libraries kicking around, so the distinction gets blurred somewhat, but there is an important difference. Most of what makes a shell script work can be used on its own; not so with a program.