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Definitively this is the type S. It's confirmed that taking a look to: http://www.autopuzzles.com/forum/index.php?topic=14833.msg135932#msg135932 there you can see the type T with its name in the license plate. In reference to the engine there are two types also been used in that car. Initially a 125 c.c. from NSU was fitted, afterwards a 2 cylinder 250 c.c. Baker & Pölling was used to drive the Brütsch Eremit cars.
According to my source, Egon Brütsch built more than version version of the Typ T. The puzzle car, with a Wendler, Reutlingen body, above is from 1950, and was powered by a 250 cc chainsaw motor from Baker & Pölling. In 1951 a envelope-body T was constructed. That car, and the puzzle car, can be seen in this un-cropped version of the puzzle picture, taken from a 1951 film clip documenting a parade held to commemorate the opening of the Prince Bernhard locks.
While you insist it is not a T, since you named the T and the engine, I'm giving you the point. I agree that the puzzle car has been shown online in other photos, identified alternately as a T and an S. Perhaps somebody with more definitive information can settle it. Either way, T or S, the point is yours.