Riot has somewhat unofficial terms, and i sort of have mine, I do agree it is becoming a vague definition
here is the type of reworks i can name
Mathmatical Reworks:

A mathmatical rework is where border line nothing about the toolkit was changed, but instead numbers were heavily swapped and re balanced to completely change how the champion scales and is meant to be played. Jarvan went from a tank to a full ad assassin, to sometimes bruiser, Shyvana went from a mostly ad based who abused her auto attacks, to a mixed damage AOE machine diver. Akali went from "one shot off Q procs" to, abuse heals with E and become tanky , Aatrox for the most part kept his identity but they put more focus on his split push, one v one capabilities while removing some of his jungling capabilities.
Minor Rework:
A minor rework is when a champion for the most part, largely keeps 70-80 percent of their orignal toolkit but is given some soft changes that are meant to widen what they are capable of, or bring more focus of what they are suppose to do. Ryze is still a machine gun aoe machine, but he was given more team purpose with his teleport and more laner capabilities with his new E. Malzahar is stillt he annoying toxic, I herp derp press R on u champ, but was given safety with his new passive, and more reliable voidlings. Shen is border line, one hundred percent the same but was given a new shield that was meant purely for basic attacks and a new Q to turn him into a more effective one v one duelist. Talon went from that awkward copy cat of zed into a effective roamer with massive amounts of mobility.
A REWORK:
A rework is when 50-80 percent of the champion is changed, but a large part of their identity remains the same. Even though Taric and poppy feela nd look completely different, to veterans who played the orignals, they will still feel at home. Taric is still a champion who stuns people, becomes tanky and heals his allies. They just changed how he does those actions now. Poppy is still that champion who hits you with a hammer while ramming you into walls, they just gave her new tricks so she isn't just a Knock on wall bot. These champions are very different in how they play, but still kept their identity.
full major rework:

These are the biggest and most challenging reworks to ask riot to do. And why they take the longest to do. This is where literally 100 percent of the champion is completely thrown in the trash can, and riot start from scratch. Riot basically hated EVERYTHING about who they were, they hated the design, they hated the lore, they hated the toolkit, EVERYTHING MUST GO. The only similarities you will see that these champions had with their orignals is that they will share the same theme, and maybe the personality will be left the same. Yorick went from a mega hyper lane bully into a split pusher who locks enemies down with his W. whose purpose becomes to be a split pushing tower destroying menis, and in team fights, he has the job of finding a target and trapping them for his team to dive on. Galio went from a "all i do is press R and nothing else" bot, to a team fighting, Teleporting shen. He now shares a lot of qualities with the shen play style in the same you farm till you are level six, and look for strong roams/teleportations the moment you hit six. You are to zone people with your taunts, and be a front line menis. He is no longer a "no ultimate, no use" champ, he is now a champion who can gank, split push, clear waves, lane bully, he has been given so much now.
Anyways those are the four type of reworks.