Playstation and X-Box support full (0-255) and limited (16-235) RGB. I can't speak for X-Box One, but PS4 can do YCbCr 4:2:0 output.
Nintendo Wii and Wii U only support YCbCr/RGB Limited (16-235 levels).
If you watch a movie on a PS3, it will NOT use what you set in the video/screen options. For movies, check the DVD/Blu ray options instead. You can enable Super White to not clip whites (16-255).
I don't trust a consoles picture viewer or their web browser, so I don't recommend making a decision based on what you see this way.
If your display has a "PC" mode, typically enabling that will ensure you get a 0-255 4:4:4 chroma signal.
To use my LG OLED E6 as an example, the PC "Icon mode" is the only way to get a 4:4:4 chroma signal. All other modes will be 4:2:0 (4:2:2? I forget since I won't use them). If you don't know what chroma is then THE WIKI
has an example showing the difference between chroma resolutions.
There is an easy way to tell which black level is correct for your display.
Set the console to full rgb. Load a game that has a black loading screen.
Make note of how "black" this is.
Switch your DISPLAY's black level to the other option, and check that loading screen again.
Is the black of one brighter than the other?
If yes, use the darker of the two on your display. Wrong option here means picture is washed out.
If both are the same, then use the one that looks brighter *for everything else. Wrong option here means picture has black crush.
You can probably do this another way if you can set your consoles background/wallpaper to be pure black. If you can make your own, make a 1920x1080 (or whatever) picture and fill it with red=16, green=16, blue=16.
Forgot to finish a sentence.