ADHD used to be called just ADD in DSM-III (released in 1980), with the subtypes ADD/H (ADD with hyperactivity) and ADD/WO (ADD without hyperactivity). However, by 1987 when DSM-III-R was published, it turned out there was not enough empirical evidence to support the multidimensional view of ADD from DSM-III, so DSM-III-R adopted a unidimensional view, which was reflected in the name change to ADHD. This change was not without criticism, and it turned out that the criticism was well founded, because by the time DSM-IV came about (it was published in 1994), enough empirical evidence had accumulated for the multidimensional view, however the dimension space was refined into inattention and hyperactivity-impulsivity, and allowed for a combined type as well. From empirical research it turned out that the DSM-III diagnoses of ADD/WO and ADD/H correspond fairly closely with
the DSM-IV diagnoses of ADHD predominantly inattentive type and combined type, respectively (cf. Morgan et al.)
Now for the terminology, the proper name howadays is attention-deficit/hyperactivity disorder (i.e. with the slash), which correctly reflects the equal importance of the two dimensions (from DSM-IV and 5) in the name. If you want to be really nitpicky, the proper, unambiguous abbreviation of this name should be something like {AD/H/AD+H}D or ADD/HD/AD+HD, though of course nobody writes it like that for reasons of parsimony.
If you propose to say that "ADD has 3 types: predominantly inattentive [...]" you're still semantically repeating inattentive which means attention deficit, so I don't get the parsimony you propose at 1 as it asymmetrical, i.e. you're still repeating something in the first subtype from the title name even though you avoid it for the 2nd subtype.
As for proposal 2 "ADHD has 3 types: predominantly inattentive, predominantly impulsive", you're now calling the hyperactive-impulsive just impulsive. This is more a matter of how do you want to call the hyperactivity-impulsivity subtype. There is probably not enough empirical evidence to to have 3 dimensions (inattention, hyperactivity and impulsivity) for ADHD, nevertheless DSM-IV distinguished between the latter two for reasons not entirely clear to me, but which I suspect are a matter of calling a spade a spade:
Hyperactivity: (a) often fidgets with hands or feet or squirms in seat
(b) often leaves seat in classroom or in other situations in which remaining seated is expected
(c) often runs about or climbs excessively in situations in which it is inappropriate (in adolescents or adults, may be limited to subjective feelings of restlessness)
(d) often has difficulty playing or engaging in leisure activities quietly
(e) is often “on the go” or often acts as if “driven by a motor”
(f) often talks excessively
Impulsivity: (g) often blurts out answers before questions have been completed
(h) often has difficulty awaiting turn
(i) often interrupts or intrudes on others (e.g., butts into conversations or games)
For diagnostic purposes these two are combined, i.e you need 6 of any of these for hyperactive-impulsive diagnosis. Nevertheless it seems somewhat awkward to say that someone who has difficulty awaiting his/her turn is hyperactive rather than impulsive. There may be more history behind this sub-subheading split without diagnostic consequences; the DSM is designed by a committee so there may have been more to it, but I wasn't able to find out. What seems obvious is that lifting hyperactive rather than impulsive to the disorder name is appropriate given that most criteria for the subtype are of the hyperactive kind. Calling it "attention-deficit/hyperactive-impulsive disorder" seems really a mouthful.
It turns out there's at least one 2005 paper that partially agrees with your criticism though:
Not only is “ADHD without hyperactivity” (ADHD of the predominantly inattentive type) an awkward locution, but it also tries to squeeze ADD into a box in which it does not belong. The term ADHD should be reserved for when hyperactivity is present (as the term implies), regardless of whether inattention is also present.
As it's evident in the above, abbreviations don't capture alternatives. So ADHD is hard to read as {AD|H}D. Nevertheless the DSM-5 committees were not convinced and the name remained what it is. Although some changes were made to the ADHD diagnosis the DSM-5:
The retention of the ADHD symptom domains and 18 core symptoms likely reflects a judgment that the DSM-IV definition of ADHD has largely withstood the test of time.
However
The change in nomenclature from “subtypes” in DSM-IV to “presentations” in DSM-5 reflects increasing evidence that symptoms are often fluid within individuals across their lifespan rather than stable traits. DSM-IV ADHD subtypes change across development due to the heterotypic continuity of symptom trajectories over time. For example, since inattention is relatively stable across development while hyperactivity/impulsivity often wane with age, many children diagnosed with ADHD, Combined eventually transition to ADHD, Predominantly Inattentive [...]. The “presentation” terminology better reflects that the symptom profile represents the person’s current symptomatology, which may change over time. The “type” terminology implied more stable, trait-like characteristics.
