Cheap media can be a problem in the burning process and in longevity. And some players/drives are better readers than others.
When you say they no longer play on other people's players, are you talking about movies you burned a long time ago, that once worked for them, or are these only new ones that you have recently converted and burned to disks?
If the dvd movies play properly in your player and in your niece's Xbox360, we can safely assume that they have been converted and authored correctly. That leaves poor burns, poor media, or burning too fast. Switching to +R media might not be the best solution unless you bitset them to look like DVDRom disks to the players. ImgBurn can do that automatically for you.
Bitsetting can increase compatiblity for picky players, especially older players that don't like burned media very much, but it is usually not a problem with modern players. If you want to try it with your +R disks, look in ImgBurn: Tools-->Drive-->Change Booktype-->Samsung tab. Change For: DVD+R Media
New Setting: DVDRom
I would try Verbatim 16x and burn at 8x speed and see if this helps. Or try changing booktype (bitsetting) with the Sony +R disks and also slow the burn speed down to 8x for them. This assumes you are using a desktop burner.