When we Regency writers include a waltz in our stories, I believe our readers see something like this — the Victorian variety, which is fairly close to how we dance it now.

In fact, it was a lot closer to this (by the 1820s):

Or even this (when couple dancing really got going after about 1810 — 30 years earlier on the continent):

Before that, the term waltz referred to a long dance to a piece of music with three beats in the bar, or to the music itself.

Then, of course, there is a modern waltz.