Idiom: einen Türken bauen. Etymology?

Question

What is the origin of the strange idiom "einen Türken bauen"?

I have already checked wiktionary, and they present a few alternatives. All these alternatices are insufficient IMO, as they all smell like someone invented them to make the idiom sound plausible.

These stories all are hard to refute, but I don't believe them.

Short overview:

Story 1:
When the "Kaiser-Wilhelm-Kanal" was inaugurated, the hymn of every State that had sent a representative was played. Because the hymn of the Turks was not available and instead the musicians played "Guter Mond, du gehst so stille durch die Abendwolken hin."

Story 2:
In the age of Rokoko Johann Wolfgang Ritter [..] built a so called "Mechanischer Türke" who won chess against some of the most renowned masters. Later it was revealed, that in fact the puppet did not think, but a human inside played.

Story 3:
Swiss Military speech: Türgg - maneuver

Story 4:
In the 15th to 17th century, there was great fear of a turkish invasion. This seems to have been used to create high taxes from nothing (to be used for a counter-invasion), which were used for something completely different then.

Answer 1

Obviously my question will not lead to a discussion of the problem. So I'm going to add my story to those of wikipedia. I learnt the expression in the German Army in the sixties where it was currently used for special effects during manoeuvres. In my old French dictionary Sachs-Villatte the French noun truc with the general meaning fraud is also indicated as a special theatre expression for machinery used to produce special effects on the stage, eg special transformations. In my view the French military took over this theatre term for their similar devices in manoevres and this term came to the German military as well. But it was transformed from truc over *Trück to Türk, and finally to Türke, a simple metathesis of the sound r. So when a German officer said "man hat einen Türken gebaut" he meant the special effect was produced by tricks with specially built machinery and devices. He did not speak of a person from Turkey or a turc who played chess.