So how about this?
This is only a naming change, it doesn’t impact functionality. So I would say, go for it!
What is the difference between these? I though they were one and the same.
Understood to be the python ‘core’?
Yes, why not if it clarifies things. 
.
It still means it is all under the Home Assistant Brand i.e ‘Home Assistant’ is not a product in itself, it is a family of products.
Well the list I made, is actually part of the list that we are implementing now.
Nope, they are not. You can install the Home Assistant Supervisor on any system you like (e.g., a Debian box), that together with the Core + add-ons becomes the “Supervised” type of setup. It is monitored and managed by the Supervisor, but not the full project. This used to be known has the Hass.io Generic Linux setup (or often miscalled Hass.io on Docker, god forbid whoever came up with that term).
Basically, yes. The Python application which used to be known as Home Assistant. Can be installed in a venv, system-wide (not recommended) or pre-packed in the docker container.
It is! If you run the full family, we call that Home Assistant.
This is where the confusion sets in “When you are a Bear of Very Little Brain” (AA Milne).
OK, I thought the Generic install gave you a full HassIO, just without HassOS (as suggested by @cogneato).
What does the generic setup lose other than the tight OS integration of HassOS?
So is the ‘Full Family’ the old HassOS or the old Hass.io?
Can you still install the ‘Full Family’ on a base OS of your choice or does the ‘Full Family’ come with the OS?
Except it appeared you could install HassIO without HassOS - this is perhaps where the misunderstanding has come from; you were not actually installing HassIO as that is not possible.
I’m not trolling, I am genuinely interested as what I though I understood, seems to be wrong and it was a major stumbling block when getting started.
This is one of the reasons for doing this whole naming thing to begin with. Yes, you can install the former named Hass.io on any (Linux based) system, you do not need to use HassOS.
Full family, in old terms (now Home Assistant) = HassOS + Hass.io Supervisor + Home Assistant Core + Hass.io DNS + Hass.io CLI + the add-ons. It uses a lot of more subprojects as well.
Os communications, single click upgrades and pretty soon a bunch of more things.
this does seem to have caused the need for a new light version - Hass.Le ? 
Upgrades work pretty well for me. I don’t notice any loss of functionality from no OS communications but that could be an “Unknown Unknown” situation.
If the result of this is to exclude the vast swathe that do not want a single image deployment, then that is a regrettable development.
Why jump into that so negative instantly? Obviously, if we know the OS, we are able to better control it. Seriously, dude, you don’t know what is, what it does, but instantly “a regrettable development”.
Just sad.
I didn’t; I said if, but your “pretty soon a bunch of more things.” and the other sentiments expressed, have come across as negativity to the ‘Generic’ install crowd and if that is a correct assessment then it would be a regrettable development and would make me sad.
Tell me that isn’t so and the development is not going to be focused on the OS integration side, thus making the ‘generic’ install a route not to be taken, then I’m sure many will be happy.
Every type of installation as pros and cons. Home Assistant does everything to integrate with anything as much as possible.
Home Assistant is a lot of letters to type.
In the process of renaming, could you also come up with official short hands?
A block diagram of how things fit together for the overarching Home Assistant vision would also greatly help.
I could see this. Maybe rename the new home assistant to “home’s assistant” and then shorten it to “ho’s ass.” That will set it apart from the previous names for sure!
One letter short of a ridiculuous name 
How about something like the following…
Code Name: hass-os
A full Operating System running Home Assistant Technologies
Official Flavors - Raspbian, CoreOS e.g. Home Assistant Operating System, Raspbian Flavour; Code Name: hass-os-raspbian
Community Flavors - Debian, Fedora
Repo: github.com/home-assistant/os
Code Name: hass-core
Repo: github.com/home-assistant/core
Code Name: hass-frontend
Frontend is independent from Core
Repo: github.com/home-assistant/frontend
Code Name: hass-supervisor
Repo: github.com/home-assistant/supervisor
Code Name: hass-cli
Repo: github.com/home-assistant/cli
Split out from the Website
Repo: github.com/home-assistant/docs
Repo: github.com/home-assistant/website
Repo: github.com/home-assistant/android
Repo: github.com/home-assistant/ios
Thanks, @sffjunkie but that is almost what we are going to do 
(except we are not Microsoft, so no code names 
)
As a two-year newbie, I can appreciate the comments regarding confusion arising from outdated forum posts, videos, blogs, how-to’s, etc. For those constantly involved with Home Assistant on a daily basis and that know the history because they grew up with it, this isn’t as much of a problem. But, I tried to be a self-starter and search and not ask questions for everything I was trying to do. I can say honestly that about half the information I found no longer worked due to breaking changes. I think any change to the names will cause some confusion. It’s unavoidable. But, changing the name of one product to an existing product will really be confusing. The example of Fred and Bob is a perfect example, and if that doesn’t move you, nothing probably will.
I am now involved with Home Assistant daily, and have an automated home with a nice looking frontend, a good understanding of Home Assistant, but it took a month of frustration and devotion to do it. I started two years ago, but let the project sit for almost two years, because I didn’t have the time or energy to devote to it when I realized how complex it was.
Notwithstanding some abbreviation confusion with HACS, I like Home Assistant Complete for the all-in-one offering. It tells the story without confusion. It’s an all-in-one solution. You get everything you need. I really dislike Core when describing what most non-developers think of as the Home Assistant application, because I think many will hear Core and think in physical terms like the center of a sphere or layers and will associate Core with the OS.
“I run applications on my Mac. The applications run on the ‘core’ MacOS.” You hear this kind of speech used quite often. Core is too often associated with operating systems and low-level systems.
With all that said, thank you to all who have spent so much of your time developing this product and continue to make it better. It is a labor of love, and I appreciate so much what you’ve made available for the community. Hopefully I’ll be able to contribute in the future.
It’s a bit late for this, but, did you guys consult with a professional about this stuff? I mean somebody who actually works with branding and the like. I say this because, as a dev, this is exactly how I would name things and at my current company we’ve been doing stuff like this and just recently hired a manager who actually understands branding and it was a real eye opener for all of us. I mean literally, the amount of things we had with “core” or “base” tacked on is very confusing to new hires or any outsiders, as they haven’t been there experiencing the evolution. It would be like “no that’s in xyz core, not in xyz base”
It’s really easy to get stuck in the dev mindset, which understandably isn’t as big of a deal here as this is mostly people who like to tinker.
Also +1 to the guy who mentioned trying to get Dash buttons to work and being frustrated over add ons. I had the same experience… I’d been running HA for I believe 2 years on docker, and hadn’t really kept up with the community as I had everything working as I liked. Then when I wanted to do dash buttons I was suddenly in this world of people saying Hass.io and it meant nothing to me, and it took honestly a while to figure it out, which happened through forum posts and not official docs. So I guess this is also me saying please make sure everything is actually updated in the documentation.
Might I remind you that often the official sources, or lack thereof, don’t get the job done.
When I was doing my setup, if it weren’t for other people/sites, combined with my tech background I’d give up.
I would add the following:
Cloud
Code name: hass-nabucasa
Repo: GitHub - NabuCasa/hass-nabucasa: Issues related to the cloud integration in Nabu Casa
The brand is a mess regarding the cloud services offered by nabucasa. The current branding is a bad one for latin languages, as it ressembles “somebody’s home that is a dumb guy”. It would be much better re-branded as a direct reference for its services or classes of, as the other ones: OS, supervisor, etc.
Home Assistant Cloud
or
Home Assistant Cloud Services
Depending if the father-branding results on a single word for its dependants or else.
Just my 2 cents.