And if there is a contract dated to the 6th October 1582, that does raise questions about its legitimacy (though in general misdating a contract does not void it, it would merely be one point of evidence in your argument that it's a forgery, just like using a particular font in a modern printed contract)
It's one of those "well technically" things programmers love to obsess over and the judge just says "everyone knows what was meant, stop being an idiot."
Yeah there are a lot of examples of dates being written two ways on the same document/artifact. In the Anglosphere there are two changes that happened close together: the movement of the new year from Lady Day to the first of January, and the Julian/Gregorian calendar jump, so that confounds things slightly. https://en.wikipedia.org/wiki/Old_Style_and_New_Style_dates