Your post makes some sense but is choppy.
In general, the booster with the green light has a couple things routed through it: The antenna and the cable/satellite outside port. You are correct that while running on antenna, the green light should be lit and the TV and radio get better reception.
When you have coax connected to the cable/satellite port, turn off the booster (green light off) and the TV will run on cable or satellite but the radio reception will likely be poor and the TV will not get antenna channels.
One thing to keep in mind is that your TV needs to be reprogrammed each and every time you make a change of location or input. Depending on what you're running on, you'll have to change the TV from "Air" to "Cable" and vice versa and then do an autoscan to find the channels that are being received.
2017 Forest River Surveyor 247BHDS...
...Tugged by a 2016 F250 XLT FX4 SuperCrew, Shortbed 6.2L, 4wd, 3.73...
...Joined with a Husky Centerline TS 800-1200lbs WDH