This is somewhat of a note to myself. Up until recently, YouTube would allow sharing videos without related videos using an option on the Share:Embed popup. I was updating some pages last night and noticed they no longer offer this option. It is still possible to keep related videos from showing - you just need to add ?rel=0 in the link.
For example, a src for the embed code has a value like this "https://www.youtube.com/embed/cIuXl7o6mAw"
Here is where the ?rel=0 (that is a zero, btw) should be entered:
Hope this helps someone else. I can't be the only one who doesn't want my students to know that I'm looking for a "cute haircut for fat ladies."