Below, I've linked a post from former L&T host Julie W on Waltz music and Country dance in JA's society. In the early C19th, I believe JA's heroines danced to Country dances.
Although the waltz as a dance was known in England by 1812, it had a scandalous reputation ,I guess, as it wasn't acceptable in polite society.
However, collections of Waltz-airs had been published in England since about 1790 and were used for Country Dance tunes and furnished with Country Dance figures and called Waltz Country dances; the dancing couples did not hold each other close so propriety was preserved, aren't we all relieved ? ;)
Below, I've linked a couple of posts from former L&T host JulieW on Waltz in JA's time. If I've (finally !) understood correctly, on the floor, people danced the familiar old fashioned figures of English Country Dances to Waltz tunes.
At the Coles party, Mrs Weston seated at piano began playing 'an irresistible waltz' (music). Frank Churchill secured Emma's hand and led her up to the top. In Julie W's opinion, the passage describes the commencement of a Country Dance.