So, how many types of Waltz did you come up with?
Depending on how you count, I got 4-6. The first two are easy: Country/Western Waltz and Viennese Waltz. Although all Waltz forms share things common to the genre, each version has its own "spin" on the dance. Country/Western Waltz has very progressive movement and is probably the next-to-fastest Waltz. The fastest? That's Viennese Waltz, which is simultaneously fast, progressive, and dizzifyingly rotational.
The other Waltzes, and the ones most commonly found in the social setting, American Waltz and International Waltz, are the ones that count as 2 or 4 styles, depending on how you look at it.
Why? At the beginning through Bronze levels of achievement, both American Waltz and International Waltz use closed footwork, like in the box step and progressives (or links) most of us know. But once you start learning the Silver level and above, both dances use what is known as continuity styling, in which your feet pass each other.
Some people look at a Bronze Waltz and a Silver Waltz and see two completely different dances. They're really not, but the advanced footwork used in Silver-and-above Waltz does make the dance look and feel very different from its Bronze predecessor.
Chew on that for a while, and I'll be back to tell you why I think each style is a valuable contribution to the Waltz genre.