Thanks, Derek!
It turns out that perhaps my issue was that I had initially edited the single, default template as the one with additional fields, because I did it before I was thinking about templates. When I went back and made the database's original template the one with no fields and created a new template with fields, I was able to achieve what I was hoping.
Now my question is, is there a way to retroactively associate a template with existing items? I see that new items in a view do show the associated template, but the existing ones do not. This isn't super important, now that I know how it works; just curious if it is possible.
Thanks,
Chris