Note: I had mostly written this response by the time you posted your next response, so this might seem a bit out of sync.
This goes back to why I wrote the blog post in the first place. When you start throwing file extensions at users, and adding in a matrix of types vs. licenses, and name both an "enterprise geodatabase" and an "enterprise license", it's no wonder that users get confused. The communication isn't clear, it's often inconsistent, and it often contradicts itself.
First, I think it's important to be able to communicate the functionality of a geodatabase without having to identify its file extension, especially when the file extension is ".sde". Like I stated in the blog post, I've started using the term ArcSDE only in reference to the ArcSDE application server, and I think it helps to create clarity. When you communicate with users using the file extension ".sde" (and I understand the historical and technical reasons for this), it equates "enterprise geodatabase" and "ArcSDE" in the user's mind. (You mentioned in your previous comment that a .gds file could be renamed to .sde -- interesting, I wasn't aware of that).
Second, on enterprise geodatabase licensing, what you've communicated here is inconsistent with the ArcGIS documentation, which identifies desktop and workgroup geodatabases as distinct from enterprise geodatabases:
From a user's perspective, these are different animals, because they're licensed differently and work differently in terms of simultaneous numbers of connections and concurrent editors (and, yes, I understand that you can take the same .mdf and .ldf files and move among the different "types" by licensing them differently). I think that the definitions that Esri provides in the above links are helpful, but are inconsistent with how Esri communicates about geodatabases elsewhere.
Finally, I think that your final sentence would be better written as "ArcGIS Desktop 10.5.1 can connect to enterprise geodatabases (versions 10.1 through 10.2.2) via the ArcSDE application server" -- again, more consistently using the terms "enterprise geodatabase" and "ArcSDE application server".
I'm not trying to be pedantic about this, but I have found that out here in the field, Esri's inconsistent communication on how users create, license, connect to, and use geodatabases causes a lot of confusion, which is what I'm trying to address in the blog post. As I indicated in my reply to Asrujit, I'll update my blog post to remove the term "ArcSDE geodatabase", and instead expand the discussion of the different methods of connecting to enterprise geodatabases.
