Rick Nelson real name was actually Eric.
The other children (actress Tracy Nelson, twin musicians Gunnar and Matthew) were over 18 at the time. Only the youngest was a minor.
Apparently, Rickey’s parents didn’t know (or care). 
The first season of Laugh-In, early 1968, had absolutely perfect music videos in every respect. Probably videos rather than film and meant to be shown on television. I bet they could have been played on MTV and nobody would have thought twice.
As I said, there are a zillion claimants, but if video and television and the band not just standing around are required, these are pretty early.
Here’s a piece of Ozzie Nelson trivia (which I’ve previously mentioned on the SDMB). When I first heard the 1968 Cass Elliot version of the song “Dream a Little Dream of Me”, I assumed that it was written especially for her. In fact, the first recording of it was in 1931 with Ozzie Nelson. He occasionally sang with his band.
Ozzie Nelson was one of the most famous graduates of Rutgers University and Rutgers Law School. I recently saw his incredibly racist cartoons that he produced for the school humor magazine. I’m sure the racism wasn’t too shocking for the 1920s.
I’ve read that Ozzie was not a musician himself and couldn’t play any musical instruments. He functioned solely as the business manager, orchestra leader and vocalist, and was quite successful at it.
I just read his wiki article, because though I knew some things about Rick Nelson because I’m a fan, I didn’t know much about his family (although I knew about the family’s TV series), and it was mentioned that Ozzie played saxophone in a band in his early career.
The first video I remember was for Foreigner’s “Cold as Ice” in episode 62 of Saturday Night Live, 1978. It featured Stacy Keach getting stabbed by a crazy woman. I was puzzled as Stacy was not singing the song, just in the video. I guess I was already behind the times.
I used to watch “Ozzie and Harriet” on TV when I was a young’un. Ricky would occasionally sing (or lip sync) a song on the show. He affected that sexy bedroom-eyed look that made the girls go wild.
Some articles say he was a proficient saxophone player before he became a bandleader. They also say he could play the banjo, ukelele, and the piano. Are there any videos of him playing any instruments?
I saw a 1940 article that said Ozzie could play any instrument in his band except the violin–which was the only one he had ever had lessons on. Now whether that’s true or not is something I can’t attest to.
Hmmm. Wish we could see a video of him playing…any instrument.
He played banjo in an episode of Ozzie and Harriet. He could have faked it, I guess, but I don’t think he would have let the other guy do the fancy stuff if he was.
He doesn’t actually play much of the time. Thanks for finding something.
According to this article:
For Ozzie Nelson, music was always an interest of his. During an interview with James Day on the Day at Night series in 1974, he remembered playing various instruments, including the banjo and saxophone,
My guess is that he was a decent instrumentalist but not as good as bunches of people he could hire. His talents shined in front of the band. So both sides are probably true.