EDIT

I put milestone to 5.19 as it belongs to #8774 content wise.

This ticket seems to be somehow contradictory to #8660. However, currently I think a menu not automatically created is preferable as it allows more content wise structuring.

My suggestion for ordering is

Feature
Character
Property
State
Structure
Structure Modifier (Modifier?)

Named Area
Named Area Level
Named Area Type
Distribution Status (use same label for term and tree editor, so not Presence Absence term as currently for vocabularies

Rank (only for tree)
NameRelationshipType (only for tree)
TaxonRelationshipType (only for tree for now, should be editable later)
NameTypeDesignationStatus (only for tree)
NomenclaturalStatusType (only for tree)
SpecimenTypeDesignationStatus (only for tree) - discuss if a common tree for both TypeDesignationStatus variants is needed
TaxonNodeAgentRelationType (should be editable but is not yet)

Language (only for trees for now)

Annotation Type
Identifier Type
Marker Type
Extension Type

Modifier
Kind of Unit
Determination Modifier
Scope

Sex

Stage
DNA Marker

====

Open term types:

DerivationEventType (not in use yet?) => if to use last section
DnaQualityType (fixed vocabulary?) => if to use last sectioni
HybridRelationshipType (currently not in use , should go to taxon sectioni with "rank"
InstitutionType (not in use) => section?
Material and MaterialOrMethod (in use already?) => if to use last section
MeasurementUnit (currently fixed, probably model update needed to open it for general use) => if to use in first section (descriptive data)
Method , same as MaterialOrMethod ?
NaturalLangaugeTerm (still needed?) ?> if to use first section
ReferenceSystem (to discuss) => if to use NamedArea section
Rights Type (in trees only) - not sure if needed => supplemental data section
SynonymType (for now not editable, maybe will be turned to an enum in future) => otherwise go to name section withs rank
TermRelationType (currently not to be used externally)
TextFormat (currently not in use)
unknown (currently not in use)

TaxonNodeAgentRelationType should be also editable for vocabularies (sorry, my comment was maybe not clear)

Language should have a separator below.

And a small change to my above description: can we put Modifier and Scope into 1 section followed by another section having Kind of Unit, Determination Modifier, Sex, Stage, DnaMarker.
The later are all Specimen related while the former are general terms.

I moved questions on unhandled term types to new ticket #9409

Some translations are missing.

Easy: Language, Extension-Type

Maybe also possible: Stage, Nametypedesignation Status, Specimentypedesignation Status, Taxonnode Agent Relation Type

Discussion needed: Character, Property (to avoid ambiguity)

translation works now as expected