As a personal surname Bagnall is by far the most common name to come down from the original place-name in the past eight hundred years (such names deriving from a place are called "toponymic").
As can be seen above in times long past the correct spelling of a name
or place was a matter of interpretation, often a guess, and has only become
standardised as literacy has improved over the ages.
As a far-flung and worldwide extended family (one estimate gives a total of in excess of 132,000 Bagnalls alone) we are fortunate to possess a unique variant of a surname which is traceable to back to one place, a tiny and relatively unspoiled village nestling on the edge of the Staffordshire Moorlands.
adoption of the second name or surnames in Europe took place from
the 11th century onwards.
When compared with families who can only relate their name to a physical characteristics of an ancestor (Little, Small, Broad), or the name of a predecessor's father (McDonnell (son of Donnell), Stephenson, Williamson), or an ancestor's occupation (Thatcher, Fletcher, Cooper, Farmer) we are lucky to be able to link our surnames to a place which we know was settled permanently about 800 years ago.
So, we're not a "clan" in the popular Scottish or Celtic tradition. In the 12th to mid-14th Century, John or Thomas or any other person who had lived in the village and moved away, probably only a short distance, would have become known as "John de Bagenhalh (or similar), i.e., John of Bagnall.
Similarly - Thomas would have become "Thomas de Baginholt"
(or similar) i.e. Thomas of Bagnall. Anyone who lived in, or moved
away from the settlement could equally have become known as, say,
John the Cooper (if he were a barrel maker) or Thomas Broad (if he
was perhaps of stocky build).