What you have described will give you a place to “test and showcase other work”, but using the “whateveryouwant.dreamhosters.com” will not assure you that it is on the same server as your existing domain. I have not checked which Dreamhost server dreamhosters.com is on, but it would be purely coincidental if it turned out to be the same machine as your domain.
As for whether or not you used the “correct process”, I’m hoping that the "firstname.lastname@example.org" was only a typo and that you did set up “whateveriwantDOTdreamhosters.com” via the Manage Domains menu (rather than just adding a mail address" or "whateveriwantAT*dreamhosters.com". If so, that should work fine except that it probably won’t be on the same server as your domain.
I suggest you repeat the Add a Domain/Sub-Domain dialogue and add a Sub-Domain to the domain you already have, such as “dev.yourdomain.tld”, test.yourdomain.tld, or demo.yourdomain.tld (you get the idea -whateveryouwant.yourdomain.tld). This will most likely get your “new” domain hostd on the same server as your original, though Dreamhost’s use of dataglobs" suggest that might not always be the case.
I wouldn’t worry too much about the “dataglobs” (you can learn more about them by searching this forum, or the wiki), but I would suggest you set your “test” domains up as sub-domains of your existing domain. It probably won’t make any real difference, but it increases your chances of being on the same server as your domain and might look a little more professional Good Luck!