Hmm. I only have one server (also with a two part name).
Regarding the two-part name, I've noticed that there are two servers (or at least two server names) connected with each MySQL: partone.dreamhost.com and parttwo.partone.dreamhost.com. It is the latter that is the host to which each of your mysql.yourdomain.tld resolves.
I haven't figured out any more than this. Anyone with additional insight should feel free to chime in.
This is purely a matter of convenience and convention. All client programs will need a hostname to connect to so you should think about naming them in relation to the domains on which their apps are running or serving. That way, your application is a bit more portable to different web hosts or accounts.
Alternatively, you can just use one name for all your domains. This has an obvious simplicity to it but may require that the name may not make that much sense in relation to the domain your application is running on.
I don't see that the databases are structurally "under" the hostname. It seems to me that the databases are independent of the hostnames. Again, the hostname is required by your client programs to access your database. They will always have access to all your databases (given that the username they are using has access to that database).
I dunno. It's certainly a big mystery to me as I'm only on one server.
I don't think this is really possible, as there is only one actual server you are dealing with with one namespace for databases and users. The hostnames are merely a convenience for you to abstract you from the dreamhost-assigned server name.
Your situation is clouded by the fact that you're on two different servers. I don't think this is usual and I'd ask support about it. Do you by any chance have multiple accounts? Or multiple panel users?
At first, I just used one single hostname. This eliminated any confusion for me and I was just happy with this situation as I just used that one hostname for all my database and all my domain applications. And then I decided to drop the domain that was the parent domain for that hostname and had to change all my web applications to use a different hostname.
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