It looks like they're gradually simplifying. They originally launched with a range of domains you could choose from: projectpath.com, clientsection.com, grouphub.com, etc. with a subdomain. Then it went to subdomains on basecamphq.com, and now just basecamp.com with no subdomains.

I remember yourname.seework.com from a previous job. As a result, all of the project managers called it Seework, not Basecamp.

That's exactly what happened. It was actually confusing for people since different people on the same project were referring to Basecamp with different names. That's one of the reasons we migrated everyone to basecamphq.com and now, basecamp.com on the new accounts.