Continuity mistake: When Biff's gang arrives at the diner for the first time, Skinhead walks behind Marty and plays with a ball. When the shot changes, he is walking behind him again, but this time he is not toying with the ball. (00:38:50)
Continuity mistake: While Marty is at Lou's Cafe, Biff and his gang enter. When Biff starts bullying George, note George's haircut with shaved sides, back (above his neck), and sideburns. In the following shots, George's haircut with its shaved areas change length. This happens again, when George is with Marty and then tries to ask Lorraine to the dance. (00:39:00)
Continuity mistake: When Marty and George are in the cafe talking about Goldie Wilson being mayor, George's head is turned facing Marty, but the next second his head is turned the other way. (00:40:35)
Continuity mistake: When Goldy (or Golby?) says, "Mayooorr", George is facing Marty. Then when Goldy starts saying, "Now that's a good idea," George is facing Goldy, without any motion of turning. (00:40:35)
Continuity mistake: As Marty watches George pedal away from the café, the amount of sunlight cast on the town square decreases greatly between shots. (00:41:00)
Continuity mistake: When Marty gets hit by the car and wakes up in Lorraine's bedroom. She constantly calls him Calvin. Marty says "Calvin, why do you keep calling me Calvin"? When he says this he puts his arm on his lap. In the next shot behind him when Lorraine replies, his hand/arm is by his side. (00:43:40)
Continuity mistake: When Marty is in 1955 observing young Uncle Joey in his playpen, the colored balls on the second rung keep changing their positions from one angle to the next. (00:45:05)
Continuity mistake: As Lorraine's family watches "The Honeymooners" around the dinner table, a round pillow on the sofa nearest the lamp repositions itself. (00:46:05 - 00:47:30)
Continuity mistake: When Doc takes the photo from Marty, the way he holds it changes between shots. (00:50:07)
Continuity mistake: When the JVC recorder is first played, the screen on the TV is a blue colour. Later, when Doc is playing the Libyan part again and again, the screen seems to be a brownish colour. (00:50:10 - 01:05:40)
Continuity mistake: In 1955, when Marty shows Doc the flux capacitor, he points the flashlight in different directions between shots (first at Doc, then the flux capacitor). (00:51:35)
Continuity mistake: When Marty is playing back the Twin Pines Mall demonstration video for Doc, the camcorder tilts itself between shots, judging by the lens which sits higher above the TV set when Doc starts shouting, "1.21 gigawatts." (00:53:00)
Continuity mistake: When Doc and Marty are looking for his dad at school, you can see a girl with a blue sweater going downstairs twice. (00:56:49)
Continuity mistake: School Cafeteria 1955, Biff groping Lorraine Baines. Between Marty trying to convince George McFly that Lorraine fancies him and pulling Biff off Lorraine, note the dis/reappearance of the child in a blue/red t-shirt and a man in a blue shirt/tank top sat at the table behind George. (00:57:25 - 00:58:45)
Continuity mistake: When George is in the school cafeteria talking to Marty about science fiction stories, he holds the notepad differently with each changing shot. (00:58:55)
Continuity mistake: When Marty is standing over his father's bed in 1955 persuading him to date Lorraine, there is a hair-dryer in Marty's belt that seems to disappear between different frames (note the hair dryer's cord leading to Marty's right hand, as he holds it behind his back). Really obvious. [This scene was originally longer, including a part where Marty removed the hair dryer from his belt. Still a mistake, but there's the explanation.] (00:59:45)
Continuity mistake: In the cafeteria, when Mr. Strickland is glaring at Marty and Biff, the table Marty was sitting at with George is behind him. When Biff leaves there is a close-up of the table and both the lunch tray and Marty's jacket have changed positions significantly from the previous shots (George had already left). (01:00:30)
Continuity mistake: When Marty pulls the family picture from his shirt pocket in front of George's house, he holds it with both hands at first. But once the angle changes, only his right hand has a grip on the picture. (01:01:45)
Continuity mistake: When Marty is standing over his father's bed in 1955, Marty puts the headphones on George and then puts the tape in the cassette player. If you notice the hairdryer is on the left side of the belt and is touching his left arm, but when they cut to Marty holding the tape player out in front of him the hairdryer is on the right side. (01:02:10)
Continuity mistake: When George goes into the cafe to tell Lorraine he is her density/destiny, the people sitting at the bar keep changing with different shots. Best example is just after George refers to his "density" - in the background there's a brunette in a blue jumper at the bar. A blonde woman and a short-haired guy then sit on her left, but then disappear in the shot of Marty sitting at the bar - on the brunette's left there's now another brunette in a pink jumper with a different guy. The brunette's got shorter hair too. In the very next shot the blonde's back with no sign of Marty. (01:02:10)
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