I got Fizbans and am looking thru it, especially for its technical jargon.
There is ambiguity because Fizbans often uses terms, like "family", in both a technical sense and in a nontechnical sense.
What is clear is:
• "Type": Dragon
• "Family": Gem
• "Kind": Amethyst
There is also the term "dragonkind", but I suspect it refers to every member of the Dragon type, since a drake is (ambiguously) probably a member of dragonkind.
Interestingly, Fizban seems to test the waters to use the term "species", straightforwardly.
Fizban never uses the term "lineage" in a technical sense, but uses it several times in the nontechnical sense (at the individual level) of grandparent, parent, child, grandchild, etcetera.
Meanwhile, the term "species" appears several times in quasi-technical contexts. For example. The deep dragon disdains all "species" that are not dragons. A dragon art object casually mentions a "species" of bear. A silver dragon personality trait refers to oneself as a member of an elder "species".
In my opinion, species is the clearest and simplest term. The distinctiveness of the term is a benefit, because it is more likely to only find usage in technical contexts. The lack of ambiguity is helpful.
To many ears, "species" sounds too modern. But Crawford has defended the term species by noting it is a normal term during the reallife Medieval Period, and is appropriate for medievalesque flavor. Fizbans seems to see if the D&D community can get used to the term species.
If the choice is between species and lineage, for the technical term, I want species.
Surprisingly, the status of dragons as a "species" remains ambiguous, whether it is one or more species/lineages - despite the entire book being about dragons!
My current hypothesis is:
The gem dragon is a species. The chromatic dragon is a species. The metallic dragon is a species. The deep dragon is a species. And so on.
However, in the case of gem, chromatic, and metallic, each species is diverse.
The gem dragon species comprises a "family" of different "kinds" of gem dragons.
I will post later if I find verbiage in Fizbans or elsewhere from WotC to confirm or disconfirm they hypothesis that gem dragon is its own species.
Corroborating the hypothesis that gem dragon is its own species, is the fact that the gem dragonborn is its own player race.
Within the context of player options, the dragonborn "races" use the technical term "ancestry".
In the Players Handbook, the term ancestry seems to refer to creature type. So where the elf is the Humanoid type, it has "Fey ancestry", where Fey is a creature type. Relatedly, the dragonborn has "draconic ancestry", probably referring to the Dragon creature type, albeit is more ambiguous as an adjective.
However, in Fizbans, the ancestry of each dragonborn race refers to a dragon family, whether chromatic, gem, or metallic. Where the family is probably a species, the technical term "ancestry" might refer to a species as well as to a type.
In the case of gem dragonborn, the "ancestry" is used in the sense of a literal "ancestor", and suggests the gem dragonborn are members of the gem dragon family. Likewise for chromatic and metallic.
The dichotomy between "monster" and "player character" holds true. But awkwardly the term monster also appears in a nontechnical sense to mean an awful menace.
Interestingly, the contrast between "monster" and "Humanoid" appears. By implication, all members of the Humanoid creature type are potentially available for player character races. Other creature types are typically unavailable for player options.
In sum, tentatively, Fizbans jargon appears to be:
type
species
species = player race + nonplayer monster
every member of Humanoid type seems potentially playable as a race
player race can comprise "variants"
family = species that comprises different "kinds"
"variants" probably refers to "variant rules" rather than different "kinds"