"App" is completely wrong, as it was introduced by Apple for user-facing programs running on a mobile phone (if Android had been the OS popularizing this, Google would have picked the first three letters of its company name as the new name for these things, and every web site would have a banner "install our goo" :-))
"App" significantly predates the iPhone. Binary/warez sites regularly used "apps" as a shorthand, NeXTSTEP used .app before Apple bought it...heck, Apple called an early Mac development framework "MacApp".
We've been calling applications "apps" for going on thirty years at a minimum.
Mobiles pre-smartphone boasted of their 'Java App Library', too. I don't know anyone who used one deliberately though, with the slow and extortionately priced data of the time.
The term "killer app" goes all the way back to the 80s. Back then, Lotus 1-2-3 was considered a killer app as was Wordperfect/Wordstar depending on your needs.
As others have pointed out, it was also used as the file extension for "application programs" on NeXTStep (.app). It continued to OS X which is where "App" came from for iOS applications. It has nothing to do with the name "Apple".
For examples look at the search hits in https://archive.org/stream/1981-03-compute-magazine/Compute_....
I wouldn't call this an application, though.
"App" is completely wrong, as it was introduced by Apple for user-facing programs running on a mobile phone (if Android had been the OS popularizing this, Google would have picked the first three letters of its company name as the new name for these things, and every web site would have a banner "install our goo" :-))