I am having a same issue. Please let me know when it resolved. Thanks in advance
Are you saying you are also making a web product that you want the videos to show in a certain way, not full screen etc? For that the answer is above.
Or are you browsing the web and prefer the videos not show full screen? This is controlled by the web site using the tags above. If they don't specify, it will default to full screen.
In theory you could contact your favorite websites and state your preference. Or contact Apple and voice your opinion for the default behavior.
In practice though I would bet most users like the full screen videos. They are easier to watch. So I would think websites and Apple will stay this way.
Perhaps try some other browsers. They may have different behavior or more customization.
Cheers.