[(Edited on 10/08/2018) Important note: I'm leaving the below-written text as it is, for any possible interest anyone might have about the consequences of such procedure that I described. But, the "solution" I've written below, ends up not being a good one, because it affects (ruins) the font rendering done by the whole system, for some fonts. The (most) correct solution, is the one that user Dai_trying
suggested, of just downloading the new version of Firefox from its own website: viewtopic.php?f=10&t=138252#p678559 + viewtopic.php?f=10&t=138252&start=15#p678679]Solved.

You were partially right,
Segfault.
I've installed Firefox Quantum (61.0) following the instructions here -
https://unix.stackexchange.com/question ... -9-stretch -
(which I was eager to have installed on my computer anyway, because of how much faster it runs JavaScript) and, with this new Firefox version, I can already watch the videos on "real.video" also (as well as browse the YouTube site
much faster).
I've checked the "Add-ons" that are run by Firefox Quantum (and compared them with the ones that are run by Firefox ESR - which I can still also run, by executing "firefox-esr" instead) and two things happened...
1) First, the "Add-ons" were still the same.
That is, on both of these Firefox versions, I was running the same "OpenH264 Video Codec"
version 1.6, and, although Firefox Quantum was still using this same codec version that had already been installed for Firefox ESR, something about the new Firefox version (61.0) made the latter now able to show the videos on "real.video".
(Maybe the way it now runs JavaScript?...)
2) Second, these changes in Firefox then caused the "OpenH264 Video Codec" to be updated for both Firefox Quantum and Firefox ESR.
And, I'm now running the new
version 1.7.1 of this codec on both - and, while Firefox Quantum still runs the videos on "real.video" fine, Firefox ESR still cannot show them.
So, again,
the difference is caused not by any new version of the "OpenH264 Video Codec", but by the new version of Firefox.
Anyway... Whatever is happening here, I've been able to solve my problem.

I'm then marking this thread as "solved".
Thank you very much for your help,
Segfault.

I just *love* the stability, much more bug-free nature, and also modular installation options, of Debian. Apart from the unfortunate adoption of "systemd" (viewtopic.php?f=20&t=129881&start=165#p671030) this distribution is *great*.