Judy, I was going to write you a book, but I think that I will go ahead and answer your questions.
I believe that I (and others) have found the records of Walter Berger/Barriger and those of several of his siblings including that of the William to which you refer.Walter Berger is shown as being baptized Wouter Berger 19 Dec 1762, son of Jacob Berger and Sarah Schneyder with Sponsors Wouter and Anganitta Schneyder and listed in the records of the Reformed Dutch Church of New Paltz, NY. (The name "Schneyder" is either a recording or transcription error with the correct name being "Sluyter" of an unknown German origin.) And, in following years, Baptisms are listed in the same New Paltz Church for Maria Berger baptized 23 Feb 1766-daughter of Jacob Berger and Sarah Sluyter and for Johannes Berger baptized 6 Jul 1770-son of Jacob Berger and Sarah Sluyter. The sponsors in Wouter's (Walter's) case are, most probably, his maternal grandparents. (The Kingston NY Reformed Dutch Church records lists the baptism of Zara (Phonetic equivalent of "Sarah"), daughter of Wouter Sluyder and Angenietjen Keyser on 17 Dec 1738.) Now, about William (Wilhelmus) Berger/Barrager, the records are numerous for the times. William's Baptism is most likely recorded in the Kingston NY Reformed Dutch Church records as Wilhelmus Berringer Baptized 12 Oct 1760 son of Jacob Berringer and Sara Sluiter (Sluyter and Sluiter are phonetic equivalents). And, there was one other known brother, Jacob.
No Connection has been found between my Walter Berger/Barriger and Jacob Berger, who married Elizbeth Finkle on 18 Apr 1791 recorded in the Lutheran Records from Sophiasburgh.
Your Walter Barringer is not connected with my Walter Berger/Barriger. His son William's birth is recorded in Anglican/Lutheran Records from Dundas County as him having been born on 3 Jan 1806 in Matilda Twp. No records are known to exist Jacob's birth, but he was born about 1798, also in Matilda Twp. Marriage records for both exist in the Liber B part of Anglican/Lutheran Record mentioned above and I have, I believe, the names of all of William's (and his wife, Mary Lexox) children (and of which my wife is a descendent) and of Jacob's (and his wife Margaret Foster) children as well.
As respects Walter's death, family tradation says that it was not a sleigh upset that killed him. It was the kick in the head by the horse pulling the sleigh that killed him.
Contact me at my e-mail address if you want more information. (You can see it by clicking on my name highlighted which follows the words "Posted by:" at the top of this message.