You're fine. As stated earlier, either tape or use a sharpie to color the white wire black so that it's obvious it's not a neutral and it is a current carrying conductor.

Depending on your environment, outdoor TVs are designed to be able to withstand high humidity and direct sunlight. They have a MUCH brighter screen and they are more robust. A normal TV used outdoors will ultimately not be bright enough unless you are able to shield it from the sun. It also may not hold up under very heavy humid climates but should do fine in dryer climates. The heat and brightness factor may impact how it fares outside. But, realistically, TVs are so cheap now, if you get 2 years out of it and it dies, just buy a new one.

I thought the same