You are correct - mouth organs can be called "garmoshka" too ("lip garmoshka" to separate it from "hand harmonika"), although if it's diatonic, these days it will go by "harp", i.e. "blues harp", at least in the circles of people who play them.
Linquistically, they are all "garmonikas". "garmoshka" and "garmon" are just derivatives.
"Bayan" is a chromatic button accordion. Traditionally with a very specific arrangement of dry MM voices tuned in unison in the treble, and an unusual (for the western accordion) arrangement of the bass reeds, where fundamental can have 4 or 5 voices, while chords have 2 voices in the same octave. Traditional bayan reeds are also somewhere between accordion and bandoneon reeds.
These days "bayan" is broadly used for any button accordion.
"Garmoshka" or "garmon" can have 1, 2, 3 rows or even 4, in extreme cases 5 (some steirische harmonikas). So the number of rows does not define what's a bayan and what's a garmon.
The instrument you are referring to has been known under different names. First, early, prototypes have started appearing right at the end of the 19th century and were called "severyanka" - "northern" harmonika. By 1930s it was already called "hromka", for example in Novoselki's "Harmonika production technology". The author states that this type is not very well known, but is starting to gain popularity with the players.
1960s literature refers to it either as "hromka" or as "two-row chromatic harmonika".
"hromka" is likely to derive from "Chromatic". Of course, it's not chromatic, but it is unisonoric. If you look at German bandoneons, you'll also see that unisonoric bandoneon is referred to a "chromatic". Generally these terms are misused quite a lot.
Nevertheless, although there have been dozens of different harmonika types designed in Soviet block countries, since the beginning, when squeezeboxes started appearing in Russia in the 19th century, until 1950s, the primary harmonika type was a "venka" - "the viennese". It is very closely related to a quint melodeon (e.g. C-F, D-G), as played everywhere in the world. Russians made them with expanded bass end, allowing comfortable playing in secondary minor home key. The best players would buy a 3-row instrument (e.g. A-D-G) with 18 basses.
Hromka has been in the background, with limited popularity in some regions.
Then, shortly after WW2, the interest in hromka exploded. One guess suggests that it is a lot easier to accompany singing on a hromka rather than a venka, and most of Soviet pop in those years has been primarily harmonically and melodically primitive patriotic stuff. Hromka layout fitted very nicely into that repertoire. Around the same time the Soviet government suddenly made the decision to axe production of diatonics and direct all production capacity towards hromkas. Even though, I remember reading that players were, in fact, asking the factories to produce more bisonorics.
Some guesses why the goverment decided to dictate the people what squeezebox they should play were:
- The government wanted to distance itself from the German roots of the "venka" after WW2;
- Switching production from multiple sets of bisonoric reeds to standardised unisonorics for all accordions, bayans and harmonikas made it easier to meet production plans;
- The Russians believe that harmonikas and bayans were invented in Russia. Including the Stradella chord bass system

. The government might have picked hromka - the type of harmonika that was actually invented in Russia - to support this myth.
Your guess is as good as anyone's, but what the soviets achieved, was erasing 150 years of bisonoric playing tradition. Gone. Almost completely wiped out by some pencil-pushing civil servant.
By 1968, Alfred Mirek writes that although Hromka has enjoyed a period of popularity, players' interest in it is rapidly falling.
At the end of 1980s and beginning of 1990s, there is a reasonably successful attempt to revive the interest in hromka, primarily via the TV programme "Play Garmon" and associated festivals. Only by this time period "garmon" and "garmoshka" starts referring to a hromka, on an unspoken and historically incorrect assumption that other harmonika types have never really been popular in Russia.
In 2024, "garmon" mainly associates with the Hromka. It's primarily a Soviet creation, really. Ideas that it is a traditionally Russian or Ukrainian instrument are based on misinformation, as these countries did not exist when hromka playing was booming. Folks dressing up in traditional national clothes to play the hromka are akin to people dressing up in medieval clothes to play a Spanish guitar. It's worth saying that a lot of effort in the 60s has been made to try and preserve the tradition by re-arranging the bisonoric trad pieces to be played on the unisonoric hromka, but from HIP perspective, the real sound of the trad pieces on bisonorics has been lost.
By the way, you won't be very popular with the Russian hromka players if you go around telling them all of the above
Me? I play Andy Cutting's "Flatworld" and Hartwin Dhoore's "Very June" on my hromka and don't really care about the dark history of totalitarian states controlling what instruments people should play, what songs to sing and thoughts to think. I just enjoy a very capable and ergonomic musical instrument.