So here's what's happened since I posted this question.
Since I am using the AdBlock Engine extension in Safari. I tried disabling it, quitting Safari, and then relaunching Safari, it seemed to work and I no longer saw the error message when playing YouTube videos.
However, I was now finding annoying ads popping up in YouTube and other web sites. So I turned AdBlock Engine back on.
Someone mentioned that Google (YouTube's owner) is trying new methods of forcing viewers to see ads whenever they try to open a YouTube video, in the hopes of defeating ad blockers. Interestingly, today I was able to watch half a dozen YouTube videos with no problems. And when I just checked to see if my Ad Block Engine was still running, to my surprise, it IS still running.
So I'm not sure WHAT happened, but my problem seems to have resolved itself.
Thanks to those of you who offered me solutions.