Looking at the "Shield" project, it seems that there may well have been two distinct Quaker Wood families arriving in West Jersey and northern Pennsylvania: a Sheffield one, and a Lancashire one. Confusion was thus inevitable. I don't know whether they were related in any way but I doubt it; it's a common surname, and they're geographically quite distance in England.
It may be worth calling that out in this project.
The following reference definitely muddies the waters. It claims that the Shield arrived in 1682, carrying the family of John Woods of ye shelter on Woodbury Brook , of Bury Parish, Lancashire. The problem is that it states John came from Sheffield! Did the author get the wrong ship name as well? Or did the Shield drop off one family of Woods in 1678 and a different, unrelated bunch in 1682?
* https://archive.org/details/sketchesoffirst00clem/page/326/mode/2up
Note that some web trees say that the boat that arrived in 1682 was the "KS Lyon". Anyone know whether this is true?