In my opinion, a reboot is when an ongoing story goes back to the beginning and has a lot of stylistic and continuity changes that contradict the original beginning. Fans often fear reboots because they think a reboot will replace the original continuity they've come to love.
Yeah, but as far as I know, the only series of that sort that has actually been called a "reboot," because that usage is so new, is
Battlestar Galactica. One example is not a universal rule. BSG has just loomed so large in the consciousness of SF fandom in recent years that it's sort of led people to divide the world into pre-BSG and post-BSG, so even though it's the only show that's been called that, people perceive that as some kind of universal, solidly established usage. If you have other examples of "back to the beginning," contradictory restarts that have been called reboots
at the time they were made, rather than described that way retroactively in the post-BSG online world, I'd like to hear them.
Some reboots immediately replace the original continuity, like the various DC Comics Crises or the Alien Nation TV series.
What?
Alien Nation? Now, I know I'm arguing that the label does not have a single narrow use, but I find it odd that you'd apply it to something like that. Yes, the TV series tweaked some continuity details from the movie it was based on, but
every TV series based on a movie does the same.
Starman retconned the events of the movie back a decade so the alien lead could have a teenage son.
Terminator: The Sarah Connor Chronicles retconned the date of T2 forward several years.
Men in Black: The Series ignored K's retirement and made L (Linda Fiorentino's character) a more veteran agent than J.
Stargate SG-1 reinterpreted the Egyptian god-aliens as snakelike parasites, changed O'Neil to O'Neill, changed Sha'uri to Sha're, replaced Creek Mountain with Cheyenne Mountain, etc. But all of those, and
Alien Nation, went with the conceit that they were continuations of the films they derived from.
Alien Nation's revisionism was subtler than in many of those other cases. It tweaked the aliens' anatomy so the makeup would be easier on a weekly-series budget and schedule, it disregarded (but did not explicitly contradict) the movie's plot points about the worker drug and the transformation it induced, it changed the spelling of Matt Sikes's name, minor things like that. Otherwise, it was overtly intended to be a direct continuation from the events of the movie, even to the point of incorporating actual footage from the movie as flashback scenes in a pilot episode in which the main characters were still dealing with the events of the movie as a recent occurrence. I don't see any way to justify calling that a reboot, certainly not in the continuity-restart sense you're using. It's a continuation with minor revisionism to adapt it to a new medium and format.
See, this is the problem with labels. Try to apply a simplistic label to too many things, and you end up obscuring the details and differences that are actually important. Labels do more to obscure what things are than to define what they are.