My desktop has my LCD monitor through Display Port and my TV through DVI. What I do is have my multiple monitors set to "Extend these displays" and make the LCD the primary display. Ive included two pictures of what my settings look like, first is with the LCD(1) selected and the second is the TV(2).
I will say that sometimes the program you are using forces the fullscreen to go to the primary screen, I have had this trouble with Everquest a number of years ago. There was a program that you could run with the game that allowed you to put the game window into boxes, different screens and a bunch of other things.
I primarily use my TV for watching movies/tv, Media Player Classic does a great job of being able to open to fullscreen in either monitor, regardless of how the monitors are setup.