Turn off hardware acceleration in firefox.
As it turned out, hardware acceleration in Firefox is disabled like this. And not through the about: config menu.
I also found out why this problem did not appear before me.
Previously, when opening training lectures in full screen mode, a separate adobe flash window worked. Which OBS captured without problems, working in fact with the adobe flash window and not with the firefox window. So, you could even open several tabs with the video stream in full screen. And record on 2 OBS simultaneously.
Now either the resource for online lectures itself (or the adobe flash working mechanism) has changed the working scheme. Well, either firefox has changed.
The problem was solved - as they wrote to me originally, it was necessary to
disable hardware acceleration.
Thank you all for your help.
PS: Sorry for the bad English through the translator.