When you open a new terminal window it shows how the
previous session connected. It's a security thing to allow you to spot if somebody is accessing your machine at a time or via a connection you weren't expecting.
When you first login, that session connects via "console", so your first terminal session will show "console" as the previous connection. However,
that terminal session will have connected via "ttys000". The next terminal window to open will therefore show "ttys000".
If you open two or more terminal sessions at the same time then subsequent "ttys" are used - "ttys001", "ttys002" and so on. Once you close a terminal session, it's "ttys" device becomes available for re-use.
Hope that's a bit clearer than mud!