After a few years I managed to find a direct male descendant of John Harrison 1720-1780 and Helen Lyth. John Harrison is the anchor for the Henry Harrison line as Henry left his farm at Greenhouses to his son John Harrison when he died in 1727. As such there is no question that John Harrison is his son.
I am grateful that this direct male descendant of John Harrison and Helen Lyth agreed to do a Y DNA test for me.
The results prove that my Joseph Harrison 1707-1780 line is not descended from Henry Harrison.
The direct male descendant of John Harrison and Helen Lyth matches the descendants of William Harrison 1715-1778 of Glaisdaleside in Egton Township.
So who were the parents of my Joseph Harrison then? The earliest records we have from 1735 and 1737 indicate that he was a mariner and ship's carpenter from Whitby. Is his ancestry from Whitby or somewhere nearby on the North Sea coast? There is also the possibility that Joseph Harrison was adopted by Henry Harrison and raised as his son but there is no documentary evidence to prove that. At least nothing has been found so far. More research is needed.