So what happens if you were using a TV to play a video game? Are you taxed for just
having a TV in your home or just using it to receive TV programs?
The latter, almost -- the license is required to watch any television transmissions 'as live' regardless of how -- so if the only TV set in the household is just used as a monitor for the Apple ][, you don't need one. Conversely you do require a license if you watch the online stream of something that is being transmitted at the same time, regardless of whether you own a TV. You also require a license to watch TV using a PC tuner, or to record transmissions with a VCR / PVR.
So the government doesn't fund the BBC, but sends vans around looking for RF
signals looking for illegal TV's? Then collects taxes to turn over to the BBC?
Not quite. The BBC pay for TV Licensing (the organisation that collects the license fee and sends the goons round to harrass people with no television) -- this of course comes out of the license fee! The charter that allows the BBC to operate has to be agreed with the government periodically, and it's within that framework that the license fee is set and imposed etc.
There were/are some functional detector vans (sometimes said to be Tempest-like so useless against LCDs I guess, sometimes said to be picking up the IF leakage which will still work), but it's likely that a lot of them are stuffed dummies. Driving a van marked "TV Detector Van" around a few residential areas for a day or two apparently often results in a sudden rush to buy licenses, whether any real enforcement action was taken or not.
However, more often than not the only enforcement is in the form of periodic threatening letters to the occupiers of addresses where no license is held (because of course everybody has a TV and anybody who says they don't is a liar!), eventually followed by a visit from an inspector -- who as I understand it has no legal right to enter the property so can be left on the doorstep! All good fun...