I got a hint from Ron Graham, that there is a stretched Stomachion version and some speculation, that this is the original Archimedes idea. By searching the web I found the page SonOfStomachion. There is an old news article showing a very similar puzzle. All this is enough to look deeper into the subject.
The term "stretched" comes from the way, how you could derive the figures from a normal Stomachion solution. By stretching one of the solutions in horizontal direction we get another set of figures. In this process quadratic cells morph to 2:1rectangles and rectangles morph either to two rectangles or to a block of four quadratic cells. In any case the base type of the puzzle is retained.
A  This picture shows how a solution of the classical Stomachion is stretched. You could possibly use another solution for stretching, but this one works fine. 

B  This construction is derived from the facsimile on the web page SonOfStomachion, where I took the picture from. The cutting of the figures (9) and (13) is slightly different than with »A«. The origin is not known, but it looks like a fairly old print. 
Whether »A« or »B« is the "real and original" variant, remains probably a mystery. From a certain point of view its also negligible. An abstraction of version »A« and »B« has the result, that the figures (1+3) and (9+13) can be merged in both cases without any impact on the number of combinatory solutions. Both versions lead to the same simplified figures, but for different reasons. The figures (1+3) are always attached because of the single atan(2/5) angle. The figures (9+13) can be merged, because with variant »A« the angle atan(1/3) has no other counterpart. In variant »B« the small triangle (13) has with adequate scaling an edge with 3*√5 length, which is incompatible with the other existing edges of 2*√5 and 4*√5 length.
There are 28150 * 4 solutions of this rectangle regardless of the simplification.
In the following only the simplified figures will be used. It's quite surprising that you can build a square form with the figures. Although they look simpler as the classical Stomachion figures at first sight, the square form has many solutions. There are also much more solutions than those stemming from permutation of the obvious local symmetries.
This picture shows one of the solutions of the square and the decomposition of the figures into the same Q,X,Y,Z cells, which had been used for the Stomachion (see [Logelium] Stomachion). A large number of half squares are on the outline and the elements with a √5edge appear only in the interior of the form. The 12 instances of the √5edges could only make a kite ring of 10 units (see also [Logelium] DiDom Kite Rings), but its actually impossible with the existing figures. Therefore all vertex and marked points have integer coordinates in all solutions.
The square with the simplified stretched figures has 3184 * 8 solutions by eightfold symmetry. A solution sample (PDF) shows the variety of local symmetries. All shown solutions are normalised by the position of the pentagon.

Parallelogram with 22732 * 2 solutions 
Parallelogram with 2606 * 2 solutions 

Parallelogram with 130 * 2 solutions 

Triangle with 50 * 2 solutions 