I’m with @beowulf on this - I always call any failure to go over a fence a refusal. If I’m talking about my jumping day in generally (show or schooling) I will say she had X number or refusals as a general statement. When I get more specific/indepth, I will clarify with either stop or run out.
I have never had anyone (trainer, clinician, fellow rider), to my recollection, specifically differentiate between refusal and run out. In my corner of the world, refusal is the umbrella term and specifically it is either a run out or stop.
About 95% of the time, my horse runs out - and it is always because she doesn’t feel she can jump it. Either I have gotten to backwards or she is in pain. It’s an extreme rarity she stops at a jump and she has never crashed through a jump. She is honest and athletic and has always refused due to something tangible, opposed to any “meh, don’t feel like it” attitude. I have since day 1 had a habit of pointing her at the jump and essentially loose rein, hang out. I do not have great depth perception so I am not able to be so exact in telling my horse when/where to jump so she’s always had to figure it out herself. That’s how I teach all horses to jump…either because I’m a super talented trainer or because of the aforementioned depth perception issue…you decide.