@LukeAlmighty program, application, software, are all the same exact things. little doodads that execute code on a device. the connotation is what separates them, where you would use them in different contexts to imply different things. the connotation could even change depending on what device you're talking about, what physical setting you're in, or who you're talking to.
@LukeAlmighty app *is* just short for application. The phrase "killer app" for instance has been in use for a very long time, in reference to software that is so good/desirable that it makes purchasing the hardware it runs on (or even doing something like downloading a different game launcher) a more enticing option. FFXVI would be a killer app for the PS5 since it was good enough for a lot of people to buy a ps5. But you could also just refer to a ps5 game as software. Saying it's a program is a little weird, as the connotation for that is something a bit more useful for say, a work environment, like photoshop or illustrator.