TL;DR: Showdown Device handling seems to count number of Showdown Devices you don't own, not the number you do. If you own some Showdown Devices, and destroy the ones you don't own via Exodian Blade (or perhaps via a Nomad planet), Showdown starts.
In a recent game, on a Maze C map, I had one AI homeworld bisect the map because there were no side branches at the right distance for it to be otherwise. I had Showdown, Exodian Blade, Zenith Trade Ship, and Fallen Spire enabled. Two of the Showdown Devices were well outside where I could reach, so I planned to ignore the showdown for this game. The Blade was reachable, so I picked a chokepoint world (and on Maze C, there are plenty of those), built all the Zenith Trader combat specials that I thought useful (Mass Drivers, Superfort, Planetary Armor Booster+Inhibitor, Radar Jammer II), and dug in. After I had the mass drivers up, I cleared everything behind that world of AI presence and proceeded through Fallen Spire. By the time I ran out of room, I had 15 Fallen Spire cities[1]. That gave me a Fallen Spire fleet sufficient to escort the Blade to the nearby AI homeworld. When the Blade destroyed the homeworld, I saw two surprising results:
(1) As noted in the TL;DR above, Showdown started. I assume this is because two Showdown Devices were on worlds I owned and the other two were on worlds destroyed by the Blade when it eradicated ~half the map.
(2) For every destroyed world, every AI-progress inducing object on the world was also destroyed, including causing AI progress and generating a Message Log entry: every Distribution Node popped; every Zenith Reserve opened; every Data Center was destroyed; etc.
[1] Along the way, I discovered that the requirement for a Spire Galactic Capital is not "build 5 spire cities", but that the cap of Galactic Capitals is "spire cities divided by 5", so with my 15 cities, I was able to have 3 capitals.
