I understand that "Discovery" is taking a different way than the Kelvin Universe but what they are both supose to have in common is a connection to the Prime Universe. The movies had classic Spock a alternate timeline that could be seen as being no different from the mirror universe or any of those universe's that Worf went to in season 7 of TNG. That means it also sprung from old continuity since old Spock goes back and time and eventually dies and Romulus blows up.
Yes, they theoretically share a common past, but the difference is in what they can do going forward. Kelvin branched off in a new direction so that it can build its own new history going forward. Its stories don't have to lead into the events of TOS or TNG or the rest, and can totally contradict everything that happens going forward from the opening moment of the first movie. Vulcan can be destroyed, Kirk can get the
Enterprise much sooner, Amanda and Pike can be killed, etc. But
Discovery is an in-continuity prequel. Its events have to be consistent with the future we know. The Sarek in this show is the same Sarek who will have a heart attack on a journey to Babel, implore Kirk to bring Spock's
katra back from Genesis, and have a mind meld with Picard as he suffers from Bendii Syndrome. All of that is in his future and he can't be written in a way that contradicts that. So the '09 movie could show Sarek reconciling with his son a decade earlier than he did in TOS, but
Discovery cannot do that, because it's supposed to be something that happened in the past of TOS. Whatever Sarek does or experiences in this show has to be something that's compatible with his life history in TOS, the movies, and TNG.
As for the term reboot I might not be using it right. I thought reboot was something that goes in a new direction but still holds a connection to the old in some form. A remake is something were you start from stratch. Some might use a reimaging but that seems like something that can describe a reboot or a remake. I think the confusion might come from the way their seems to be endless ways to describe something. We could use retooling or prequel or updating.
Well, it depends. As an industry-insider usage, "reboot" means any revival of a dormant media property. But in fan usage, ever since the 2004
Battlestar Galactica reboot, it's been assumed to mean something like that show, a completely reinvented version that's out of continuity with what came before. By that definition, for instance,
Spider-Man: Homecoming is a reboot because it starts over and ignores the previous Spider-Man movies, but
Fantastic Beasts and Where to Find Them is not a reboot because it's in the same continuity as
Harry Potter. The new
MacGyver is a reboot because it starts over completely with a new version of the character, but the new
Doctor Who is not a reboot because it's meant to be a continuation of the original series. Then you have things like the Kelvin films and the past few X-Men films, which use time travel to have it both ways -- they're part of the same narrative continuity but they alter its events, so it's effectively a reboot while still being a continuation.
So anyway, by that terminology,
Discovery is not a reboot, it's just another prequel series like
Enterprise was.