I just came across your information and was interested in what you had posted.Do you have information on the line of Beat which you believe is the brother to your Peter?i see that your line has kept the traditional name with the "y" while my husbands line has went to the "i".My husband is a 12th generation descendent of Beat Rhyner.My information shows that Beat was also born in the 1570's.
Your pictures are great.
So far I have found 3 variation of the spelling. rhyner, rhiner, and rihner.all starting from the early 1570's.
I would be happy to share information with you if you are interested.