Other mistake: When the Libyans shoot at Doc Brown and the bullets hit his van they leave no bullet holes, just impact sparks. (00:29:10)
Other mistake: The car that almost runs Marty over has the vehicle registration plate of 6S 48405. Later on, when Marty is hit by Sam, Sam's car has the same plate. (00:35:49 - 00:42:02)
Other mistake: The first time-travel experiment involving Einstein the dog, he goes one minute into the future after the DeLorean excelled to 88 mph. Yet after the minute is up and Einstein returns, the car emerges already braking, and screeches straight to a halt. At 88 mph, it would have to travel much further even to quickly stop the car. Look at all other time-changing moments in the three films. The car takes much longer to stop.
Other mistake: When young George and Marty are in the backyard talking about what will happen at the dance, young George hangs up clothes to let them dry, but if you look closely the clothes are actually already dry.
Answer: The video camera was in the DeLorean. With the right kind of adapter, which was common enough in the 80s that Doc might've had it on the camera or been able to jury-rig something in the 50s, it would have been possible to connect it into the antenna screws in the back of the TV like an old Atari and play it directly from the camera.
Captain Defenestrator
TVs in the 50s had a two prong antennae connection (two screws in the back that you put a prong antennae into) TVs in the mid 80s also had this. The coax connection (the one wire that screws in) was starting to become common, but, the two prong connection would have been more likely on any given TV at the time, so, whatever wire they used to preview recordings probably had that. very convenient that Marty brought those cords with him.
An old Atari 2600 RF Adapter would be how one would link a video camera to an old-fashioned television. A simple-enough part that Doc could probably make one with 1950s technology.
Captain Defenestrator