There is a very easy answer for this - thought I even mentioned it before. Some time ago Youtube began offering higher quality versions of the videos. There is a little caveat to this. The original uploaded video must be of high quality. I have found high quality mpeg 4 works well for this. When you first view a video directly on Youtube it defaults to low quality. If the video is available in high quality you will see some words like "watch in high quality" under the video on the right side (see pic below). Click that and it will reload and play the video in high quality.
Now, when you embed Youtube videos here on the forum they will default and play in high quality - if it is available that way. This was one of the big reasons I put that new embed video addon here a few weeks ago. Hope that clears things up.