Regarding certain games causing strain/headaches I've only come across one on PS3 which was Yakuza 3.
The thing is this was 100% fine on my old 2007 CCFL backlit TV and a plasma TV but it's painful as hell on my much newer 65" 4k panel which is also the only LED backlit TV I have ever come across that doesn't cause me any symptoms. Every other PS3 game has been fine on that TV as well.
It seems the monitor/tv you're using can also play a roll in games causing you eye strain or headaches. Maybe since it's a console game and they don't always run at native resolution it is a mix of the resolution and aliasing + the panel meshing together in a bad way for us?
Regarding different consoles all PSP versions and Nintendo DS/DSi have been LED backlit and I'm fine with them but I just can not use new LED backlit products. Original 3DS was mildly straining the New 3DS is one of the worst triggers I've had. Vita OLED was 100% fine and I haven't tried the normal LCD Vita slim
Those of you who have used a psp 1/2/3000/go or DS/DSi did you find you were okay with those?