I would also like to vote for this feature. While getting it directly implemented would be ideal, it can as an alternative, at least be added as a votable feature.
It turns out this is not always true and it is not about on which monitor was the player started. It has something to do with maximizing the window. I believe it was started not maximized and on whatever monitor you maximize the window, it goes back to it when you go full screen.
I support this feature request! I was just about to open a new issue about w3m not complying with the XDG Base Directory Specification, when I saw this feature request exists.
I can't find the option to edit the ticket, so: Where it says "if you start SMPlayer to another screen than the one it started on" should be "if you move SMPlayer to another screen than the one it started on"
Fullscreen returns player to default screen