Like so many things in history, the exact point at which "Swing Dancing" began is hard to pin down, and depends much upon your definitions, keeping in mind that the people of time might not have necessarily shared your terminology or organizational schema.
As often happens, "Swing" music came first, and it took dancers a few years to decide that what was being done to that music constituted "Swing" dancing.
Here's my understanding of the sequence of things.
In Harlem in the late '20s and early '30s musicians like Fletcher Henderson and Duke Ellington were using the expression "Swing" to refer to a certain feel in their particular style of Harlem jazz music (a style the white audiences often called "Jungle Music"). Ellington's "It Don't Mean a Thing if it Ain't Got that Swing" dates from 1931, but the idea of describing what one might dance to it as "Swing dancing" does not seem to have existed yet. Meanwhile jazz musicians, to include a few white musicians like Benny Goodman, were paying attention and were embracing this new Swing style, even if the mainstream hadn't quite caught on.
The arrival of the term "Swing Music" in white mainstream popular culture can be pretty much traced to August 21, 1935, when Benny Goodman's radio broadcast of his gig at the Palomar Ballroom in Los Angeles sparked a nationwide sensation that lasted, more or less, for a decade. From that day forward, the music of America's youth was Swing Music, and the dances they did to it were, therefore, Swing Dancing.
If the preceding pages have shown anything, I hope it is that Swing did not spring, fully formed from a void. The youth of America did not cast aside the Minuet and the Gavotte to embrace Swing. They were already doing some pretty energetic, jazzy dancing, but combining their existing dances with Swing music, some new Savoy ballroom moves, and a certain Harlem Hep Cat sensibility, imbued them with a new drive and energy and formed the core of a youth culture that was stronger and more vibrant than this nation had ever seen before. This was magnified by Hollywood, which embraced Swing and placed it front and center in countless films, spreading it to parts of the country that would have never heard of it otherwise.
Swing dancing, coming from multiple sources, was not a cohesive and uniform whole. It had multiple simultaneous currents and like everything that came before it, was a bit of a muddle.
The phenomenal popularity of Swing brought with it an awareness of its Harlem roots, and one of the main drivers of Swing dancing, as it existed in 1935, was the dancing of the black kids at Harlem's Savoy Ballroom. They created a style of dance in the late '20s they called the "Lindy Hop" or just "Hop", which was fully formed by the time Swing Music became mainstream, and the performances of "Whitey's Lindy Hoppers" on screen and in public performances; and especially their influence on the white kids who came up to Harlem to the Savoy, spread the highly kinetic style that quickly became known as "Jitterbug".
The other main parallel strain, that had its roots in the Charleston-Collegiate styles of white kids in the early '30s was what came to be known as "Shag" (a term that emerged some time in the mid-'30s). It typically included a lot of fast footwork, often with a close body-to-body hold (see the kids with their hands in the air in the photo above)
The two strains were not exclusive. While each region, group and even school might have it's own distinctive stylistic elements or signature moves, there were few examples "pure" dance of any style, with dancers constantly stealing/borrowing from each other any move that looked good to them - and since most kids were just copying what they learned from other kids or saw in the movies, it was mostly an energetic, chaotic mess.
I have a personal example of this muddle. I made the statement at a presentation that I had not found any visual evidence of the Balboa style outside Southern California (see video below). Afterwards, a man came up to me and said that everyone at his father's school in Minnesota danced the Balboa - only with different steps. I might have responded "If you dance what everyone else calls Shag, and call it Balboa, is it Balboa?", but just left it at "Oh, that's interesting".
Here are a few videos, of the scores available on YouTube, that show some of the various flavors of Swing dancing. I am only lightly skimming the surface of what there is to see and know about this topic.