Continuity mistake: George McFly's holding a pen in his right hand, then a cut to a wide shot reveals him not holding a pen as he rests his arm away from his body on the table. Then when the camera cuts back his arm is back in front of him holding the pen. (00:14:10)
Continuity mistake: At the beginning of the movie after Biff leaves the McFly home. There is a scene where Marty and George are talking. In one shot George has three pens in his pocket, a few seconds later he has two. (00:14:14)
Continuity mistake: After Lorraine drops Uncle Joey's cake on the kitchen table, Dave can be seen resting both hands on the back of his chair. When the shot changes, both of his hands are elsewhere. (00:15:00)
Continuity mistake: When the McFlys are together in their kitchen, Lorraine asks everyone to drop Uncle Joey a line. At this point George lowers his left hand and seemingly reaches for silverware on the table. But between shots, he lifts his hand back to its original position all of a sudden. (00:15:00)
Continuity mistake: During dinner, when old-Lorraine says "We all make mistakes in life, children", she swaps from having one hand on the glass to both hands, between one frame and the other. (00:15:06)
Continuity mistake: Beginning of film. When Marty arrives at Twin Pines as the Doc requested he cruises up alongside Einstein on his skateboard, saying "Where's the doc, boy?" There's a cut to the truck opening, then back to Marty, and for some strange reason the yellow wheels on his skateboard have turned pink, but the reflection on the wet car park is still yellow. It cuts back to the DeLorean coming out of the truck then back to Marty, and his wheels are yellow again. This doesn't happen in the VHS version. (00:17:40)
Continuity mistake: When Marty rides to the mall he is riding his skateboard, but after Doc comes out in the car, it's gone, then at the end of the film when Doc drops off Marty he has it again. (00:18:40)
Continuity mistake: When Doc and Marty are at the Twin Pines Mall and Doc is telling Marty about the car, he says "never mind that now, never mind that now..." and Marty has the camera facing the ground, but later when Doc in 1955 is watching the video, you can see him saying that on the video. (00:18:50 - 00:50:05)
Continuity mistake: When Marty arrives at the Twin Pines Mall at the beginning of the film, he drops his skateboard behind the DeLorean. When Doc starts the car with Einstein in it, the skateboard disappears. (00:19:05)
Continuity mistake: In one shot of Einstein strapped in the DeLorean, if you look closely in between the steering wheel, the yellow time circuits on the bottom read the month as "10" rather than "Oct." And the red and green time circuits read "11" instead of "Nov." (00:19:20)
Revealing mistake: When Doc starts up the DeLorean watch the remote when he says to Marty "not me, the car, the car." The counter is static at 00 but the car is moving. (00:19:40)
Continuity mistake: At the beginning of the film, when sending the DeLorean into the future, the wide shot shows the DeLorean heading for Marty and Doc then disappears leaving a trail of fire. Notice the cut as the car park goes from being dry to wet with puddles. (00:20:00)
Continuity mistake: When Doc points out that Einstein's watch is in sync with his, the LED display changes from being off in a wide shot to on in the closeup. (00:20:05)
Continuity mistake: Before Doc's time travel demonstration, he shows Marty two synchronized clocks. Here we see Einstein turn his head towards the front of the DeLorean, but in the next frame he's facing the camera instead. (00:20:05)
Revealing mistake: When Doc is using a remote control to drive the DeLorean, they show a close-up of the controller. The controller's power meter (the dial with a needle in the middle) shows no power - it's right over to the left. How's he driving with no battery? (00:20:15)
Visible crew/equipment: In the scene in the beginning where Marty meets up with Doc for the first time, Doc shows Marty how the DeLorean works with the remote control with Einstein in the car. In the shot that switches to see Doc put the car in reverse on the remote control, you can clearly see a crew member manually steering the car in a dog costume (You can even see his hands turning the steering wheel). (00:20:30)
Continuity mistake: When we first see the machine in action, the travel is shown from three different angles. In the second, a line of fire passes directly between Doc's legs, but in the third angle, they pass to the left of both of his legs. (00:21:00)
Continuity mistake: Moments before Doc and Marty witness the DeLorean's first trip through time, Doc's left hand changes its position on the radio control unit between shots. Initially he's holding a toggle switch, but after saying, "You're gonna see some serious s***," he has a grip on the unit's side instead. (00:21:05)
Revealing mistake: After Doc sends Einstein back in time, it shows fire trails on the pavement. Marty's leg is right in a (superimposed) flame, with no ill effects at all. (00:21:05)
Continuity mistake: When Doc and Marty are watching the DeLorean burn out from across the vacant Twin Pines Mall parking lot, the radio control unit's antenna is seen lowered one shot then fully raised in the next, despite Doc not having touched it. (00:21:20)
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