There's a few variations. Words, sayings, and useage change over time.
Short version: A stool pigeon is also a decoy, and Kid Creole and the coconuts made the term fashionable as it's usage as an informer for a time in the 1980's, but it's usage very much predates that.
Long version: It first formally appeared with both words together around 1820 as 'stale pigeon', with stale believed to derive from the Frence 'estale' - a decoy that was used to entice hawks into a net. And 'pigeon' a term for a gullible individual who believed whatever they where told.
(there is an almost certainly incorrect etymology regarding a passenger pigeon hunting practice in the US, combined with the 16thC imported English word 'stoale' for a tree stump, claimed in the US in the 1870's)
But the two words appearing side by side thoroughly predate the phrase.
In the 15thC it's in the English language as a 'stale', a creature who baits or entraps another' as in Shakespeare’s The Tempest, when Prospero demands of Ariel:
“The trumpery in my house, go bring it hither for stale to catch these thieves.”
An alternative spelling was 'stall' appearing in records to describe the decoy who distracts, so that the pickpocket can get in to rob the target. Actually still in use with 'stall' or 'stalling for time.'
The decoy got exactly the same sentence as the pickpocket, yet far less of the spoils, and is thus described as a 'pickpockets pigeon' - a foolish or gullible person already, but in this usage one that becomes a decoy.
The decoy often turned on their pickpocket to save themselves when caught, and by 1840 is being referred to as a 'stale pigeon', which starts to be spelt as 'stool pigeon' and the verbs 'stool' and 'stooling' are in use for treacherous informing, as well as 'stall' and 'stalling' for decoying behaviors.
By the 1920's it's become the slang word 'stoolie' for an informer, and a 'stool pigeon' in America, is entirely used other than in hunting circles, to describe a criminal informer, but in England it is still in use as both, especially during WW1 meaning either a decoy or informer.
In the 1960's 'stool pigeon' is in use mainly only appearing in slang, as either an informer, or for a male using decoy tactics to put women on the back foot defending themselves. (argued as to if this emerged from feminists, or prostitutes)
The chess board saying with 'stool pigeon' as don't bother arguing with a 'foolish decoy', was in use during the 1972 Boris Fischer vs Spassky chess tournaments, and is believed to have emerged from Russian Jewish diaspora circles when Fischer foolishly chose to use antisemitism as decoying behavior to deflect any possible notice of his Jewish ancestry and was known for eccentric behaviors, including allegedly sweeping the pieces from the board, thus not worth playing with, regardless of his popularity.