You are probably right but can someone answer this for me? Isn't Lancaster part of Lancashire and York part of Yorkshire? In Pennsylvania, the town of Lancaster is the center of Lancaster County and York is the center of York Counties. Both counties are last areas of suburbs, tiny towns, farms, and woods. So to me, there "might" be a correlation with the city and county relationships in both countries.
I am not sure of your point and fear you missed mine. I was commenting on your original suggestions that Lancaster, Pennsylvania is named after Lancashire and York is named after Yorkshire. My point is that the cities, Lancaster and York came first by some centuries. So Lancaster gave its name to Lancashire and York gave its name to Yorkshire. In a sense, Lancashire means The County of Lancaster, and Yorkshire means the County of York. We have at least one county where this relationship is made very plain. We very, very rarely, if ever say Durhamshire. It is County Durham viz. County of Durham.
Yes, Lancaster is in Lancashire and is part of Lancashire in the sense that it is the County Town (administrative headquarters) of Lancashire and geographically. In contrast, York is geographically in Yorkshire and can be said to be geographically part of Yorkshire but as far as local government is concerned it is a separate unitary authority.
So, I reiterate, Lancaster, Pennsylvania is named after Lancaster, England and likewise, York Pennsylvania after the city of York.