Well, for me reboot means you have the same pieces but reset and restarted. And Wikipedia seems to back me up on that. "A reboot usually discards continuity to re-create its characters, plotlines and backstory from the beginning". Although I concede it says "usually".
Wikipedia is for common use and publicly edited, so it would favor common usage. My point is that industry insiders have routinely used it differently than laypeople. I've seen plenty of cases over the past couple of decades of industry professionals using "reboot" interchangeably for continuations and reinventions, while I've only seen laypeople insist it can only be used for reinventions.
After all, to industry insiders, what matters is the intellectual property and whether it's an active production that employs people. Reviving a dormant property means new opportunities for work and profit, regardless of whether it's a continuation or a reinvention, so the continuity consideration is of secondary importance to them. But laypeople have no creative, financial, or practical stake in the property; their focus is on the story itself. So to them, the in-story considerations like continuity are of more central importance. So each group uses the term in a way that reflects their distinct perspectives and priorities. As with many vernacular terms, it's used differently by different groups, so there's no sense in arguing which is the "correct" one.
I think though we've had a word for a long time that means something that follows on from the original - and that's a sequel. Or a spin off.
Except "reboot" differs from those terms in that it usually applies to the revival of an old property after a significant passage of time. Again, to industry insiders who see it as a job or a profit opportunity rather than simply a story to watch, the abstract question of internal continuity is not as important as the pragmatic question of whether the production is
active or not.
TNG was a sequel to TOS as it had some of the same people and bits, but things progress.
Although Roddenberry approached it as what we'd call a soft reboot, intending to ignore those parts of TOS and the movies that he wasn't happy with or that were too outdated and make TNG a reinterpretation (e.g. replacing the 1990s Eugenics Wars with the mid-21st century WWIII and Post-Atomic Horror, and pretty much ignoring all the androids encountered in TOS and treating Data as unique). But his successors included TOS fans like Ron Moore, and they brought TNG's continuity into closer alignment with TOS's.
In general, historically, creators have been far less purist or absolute about continuity than fans have been. After all, nobody is as aware of the illusory nature of fiction as the ones who create the illusion. They invented it to begin with, so they know they can reinvent it however they wish, and just pretend it's continuous with what came before. Creators these days have become somewhat more digilent about continuity in response to fans' preoccupation with it, but even today there's still a fair degree of flexibility in revivals' approach to past continuity, as seen in things like
Strange New Worlds and
Tron: Legacy (which pretends to be a direct sequel but re- or misinterprets a lot of things from the original). So the dividing line between a continuation and a reinvention is not as sharp as fans tend to think. As with most things in life, there's a lot of gray in the middle.
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