The # 1 in your list illegal game circa "mid 1950's - mid 1980's" pieces that do not bear any name at all would be safe to call them simply as "Aspinall" in honor of John Aspinall - the owner of the Aspinalls casinos illegal and the legal in London and elsewhere at the UK. Would this be "the legal history" of these pieces without markings?

Considering that you will go down in history as the legit oral historian in these particular chips - the group #1. You as a reliable researcher and a first class chip collector and a dealer found the historic source from one of the family of John Aspinall, has all the rights to define this piece of history as true and correct to the best of your knowledge lay claim to name these chips now "for the record".

How would you name these finds - Aspinall, Aspinalls or Aspinall's - the "Aspinall" is to honor John the founder and the originator, "Aspinalls" as the definition of the conglomerates (illegals and the legals as a group - plural) or Aspinall's - the possessive.

You are right when you said that it does not matter which name you use - their usage depends on the locations of the casinos.
These "no name" chips will go a long way in each and every chip collectors databases - one maybe will use Aspinall or Aspinalls or Aspinall's or interchangeably or all. Sir David Spragg look down after one or several "historical epoch(s)" - no, no, no children listen the "no name" chips name is no name, T H E E N D!

What think you so?