Yes Tom, the satellites of Jupiter (Io, Ganymede, Callisto and Europa) do pass in front of Jupiter. I mentioned about transits and shadow transits. These are a well documented phenomenon and you can find predictions of their occurrence in many astronomical journals. These are possible because the satellites are much smaller than Jupiter and so appear star like through telescopes. Against the disk of Jupiter they can appear like pale dots with a dark shadow either following them or preceding them during a shadow transit.
What cannot happen, for what I would hope are obvious reasons to you, is a star being visible through the Moon. You can post as many extracts from as many books as you wish (preferably more recently than the mid 19th century). All are subjective to what the observer perceived at the time and I can tell you now that a lot of observational accounts from the 19th century and earlier are not entirely accurate down to the relatively poor quality of optics that was available then. Galileo for example saw that there was something unusual about Saturn when he observed it through his telescopes but he could not tell that he was seeing rings around the planet.
When the Moon is a cresent you very often see the outline of the full Moon very faintly illuminated. You have probably noticed it yourself. We call it Earthshine because it is due to sunlight reflected off the Earth onto the part of the Moons surface which is in darkness. Since the Moon is moving east with respect to the stars, when the Moon is at waxing cresent phase just after new Moon, you will see the star disappear behind the dark part of the Moons disk. If it happens to be a grazing occultation with initial disappearance on the dark side of the Moons disk, as the star pops in and out of ravenes and valleys against an invisible or nearly invisible limb, it may well give the impression that the star is actually shining through the Moon. That occured to me in the first extract you posted.
As for the videos posted, the first shows a very typical gazing occultation which is not uncommon for Aldebaran. As for the second one. No idea what the blinking white spec is. Probably a small internal reflection going on somewhere in the optical system. It definitely isn't a star though. I didn't think FES took much notice of videos/photos anyway for the reasons stated in the evidence page. For an experienced observer it is quite easy to tell which ones have been faked and which are genuine.