I believe the tokens are classed as "Lancaster" because the most common inscribed edge variant for this token is "PAYABLE IN LANCASTER LONDON OR BRISTOL".
I would suspect that such tokens were produced by the British token-makers, with the intent of exporting them to the US. This was still prior to the Coinage Act of 1792, so the US has as yet no official coinage nor national mint, and the various states were still making do with whatever coins or tokens they could scrounge up, and their was profit to be made by foreign private mints selling cheap underweight tokens to desperate Americans.
It may be worth pointing out that "Unanimity is the strength of society" was a motto of the
Antient and Noble Order of Bucks, a Freemason-style fraternity based in Birmingham (a token manufacturing centre) in the 1700s and 1800s. So I suspect a member of that Lodge may have been involved in the design of the tokens.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis