I would recommend contacting Facebook for your issue.
I've tried contacting Facebook and got their usual non-committal response:
Your feedback will be used to improve Facebook. Thanks for taking the time to make a report.
Even one of my friends who works in IT is stumped by it.
It's ultimately going to come down to the particular video and how it was embedded, and the operating system and browser that you're using. This is ultimately a problem that Facebook needs to fix, and there's not enough detail here to guess about possible workarounds. It's really odd that you see it work in the same browser in multiple profiles, but it's totally normal for large services to run experiments on a small subset of user accounts. You may just be seeing the outcome of an experiment in one of your profiles.
Thanks for your reply and input which I really appreciate.
My operating system is Windows 7 Home Premium,Google Chrome version 61.0.3163.100 and is up to date.
This happens to all videos I post from Youtube and to make it even more confusing,I have found that my wife's Facebook profile has the same issue.
But,as you said,I think it is a Facebook issue and it is strange that my two back up profiles are not affected although they are newer.
I am leaving this open in case I get a response from Facebook and will put an update here.
Do you have any extensions installed that might account for it? AdBlockers, etc?
I do have Adblockplus extension but this was installed way before this issue started and the other extensions I use are McAfee WebAdvisor and SiteAdvisor and these have been temporarily disabled but the problem persists.