In Craig Marks and Rob Tannenbaum's 2011 oral history of MTV, I Want My MTV: The Uncensored Story Of The Music Video Revolution, there are quotes from Bob Pittman and Steve Casey, MTV executives involved in programming, about the launch of the station.

Bob Pittmann: We launched on August 1, 1981. Our first video was the Buggles' "Video Killed The Radio Star". It was too obvious not to do it. When you see the video today, you go, "Don't tell me that's cutting edge." But at the time, it was.

Steve Casey: Nobody wanted to launch with "Video Killed The Radio Star." They thought we had to play a hit. I said, "Nobody's going to be watching. It's symbolic." In fact, the second video we played was symbolic too: Pat Benatar's "You Better Run" was a message to the record labels.

When Steve Casey says that "nobody's going to be watching", part of the back story to this is that, when MTV launched, it was not watched by a large amount of people. According to the Marks & Tannenbaum book, the majority of cable operators in the US were reluctant to carry the station, because they were concerned that rock videos would be controversial in the often rural areas they serviced. It was only after the 'I Want My MTV' ad campaign in mid-1982, with pop stars saying 'I want my MTV' in ads that most cable operators started to carry MTV. So yes, the humour of playing that song first was the point.

And in regards to your uncertainty over the video, the first video is the correct one - for what it's worth, there's a higher quality official version on YouTube that I can access here in Australia but which may not be accessible in your country - I'd guess that the second video was footage of the band lipsyncing on a German television program called Disco. In regards to the "Video Killed The Radio Star" video, the Marks & Tannenbaum book has this anecdote from its director Russell Mulcahy:

Russell Mulcahy, director: "Video Killed The Radio Star" was a one-day shoot in South London. I had a good friend who was trying to break into acting. I told her I needed a girl to dress in a silver costume and be lowered via wires into a test tube. What she didn't know was that we would need about thirty takes. What you see in that scene is actually the wrong take; the tube falls over, which wasn't supposed to be in the final cut. I think it's still in the video to this day. No one really knew what they were doing, including me.

I do recommend the Marks & Tannenbaum book for what it is - there are some hilarious anecdotes of 1980s excess.