LOL....I'm not sure that I am!
Just trying to understand whats going on.....
From looking at it again the site with the icons should provide you with icons that look like the ones in their header image.....with, I now notice, a drop shadow.
You're saying that when you generate the icons you want on that site and download them, when used in the app they have that white background to them.
If this is the case then nothing you do in PS is going to change anything, the icons are, as far as I can see, perfectly ok for use in the app without having to alter them in PS first.....or so their header image would lead you to believe.
As to why they don't appear the way they should in that app is, to be honest, a mystery.
I don't believe this is a transparency issue that should be solved in PS.
Its a problem with the app, the icons or both.
Your best bet is probably to find a forum dedicated to that app and see if anyone else has had the same issue with icons from that site.
Sorry to make you redo stuff unnecessarily, I should have looked at the problem more closely than just answer the question.
Good luck anyway.
Regards.
MrTom.