Quote: "You have to rename it"
No, you don't.
Quote: "change the icon with one of the common resource manager"
Yes, you do.
Quote: "I did see a file called AGK.exe, but the date of it was last month. But I deleted it and recompiled and it reappeared, but with the old date."
The .exe file is a common file that gets copied over from the ...\steamapps\common\App Game Kit 2\Tier 1\Compiler\interpreters\ folder so they will ALL have the same timestamp. Every AGk app uses the same .exe file, it's the emdia and .byc file that change.