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Doesn't the astrometry.net plate solver output the classic "plate
constants"? With them you can transform the image's xy pixel coordinates
(measured with the Windows Paint utility or similar program) to the
corresponding right ascension and declination.

From the downloadable FITS-files provided with the analyzed photo I was only able to look at the one called "new-image.fits" using the program DS9. This program shows then the analyzed photo and for the cursor location both, the x/y-pixel location and also the corresponding RA/Dec from the fitted function.

As part of his off-line help to solve my "problem", David Walden provided me also with the following publication

"The SIP Convention for Representing Distortion in FITS Image Headers" Shupe and Hook, Astronomical Data Analysis and Systems XIV, ASP Conf series, vol XXX 2005 ( http://irsa.ipac.caltech.edu/data/SPITZER/docs/files/spitzer/shupeADASS.pdf )
 
which explains the type of function used by astrometry.net for their fits. If I remember right the classic 6 "plate-constants" which you mention do only depend linearly from x and y whereas their 6 parameters seem also to contain quadratic terms.

Marcel