Matarilevich: Gothic Checkers is not a new game, according to this article it has been around since about 1650. It also goes on to say that Turkish checkers is an early varient of Gothic checkers and that it's possible the early version of Turkish checkers had the same rule as Gothic checkers about the king only being able to move one square. Later changed to the queen with the addition of long range moves.
According to Ralf Gering, Gothic Checkers is actually known as "Altdeutsche Dame" (or in Old German:"Damm-Spel"). He also says: The old books don't say how far the kings are permitted to move. A new book by R. F. Müller which is quite reliable as judged by the other rules it gives (of draughts variants and Salta) says that that the king may only move one square (so you would be correct!). Another book which is even newer claims the same but the author used obviously Müller's book as his source. The strange thing is that Altdeutsche Dame obviously influenced Turkish Draughts (which was invented shortly after the Turks tried to conquer the capital of the Holy Roman Empire of the German Nation, Vienna, in 1683) which has long-ranging kings. Did the Turks invent the long range independently? The France introduced the long-ranging king's move in about 1650 and the new rule rapidly spread to Germany. Perhaps the early variant of Altdeutsche Dame had indeed the short move (as explained by you, i.e. the move of the modern Chess King), but later adopted the long range (i.e. the move of the modern Chess Queen). Another strange thing is that the old game books state that you must choose the move which captures most enemy pieces. This is a rather modern rule usually associated with the long-range. A mixture of "old" and "new" rules (if the short move is correct) would mean that Altdeutsche Dame was invented around 1650, probably in what is now Austria near the Turkish border. It appears that the rules are not completely known from the old sources as a German draughts expert I met 15 years ago claimed that the rules are "RE-constructed.