Huntington builds the real field from S more or less as
modern logic texts do. This doesn't quite amount to a
categoricity proof for the axioms for a complete ordered
field: you need to show there's essentially only one way to
construct the reals from the positive integers. Who
finally stated the theorem that way?
The theorem that any two complete ordered fields are
isomorphic is an obvious consequence of what, in my note
from April 23rd, I called Bettazzi's structure theorems:
any two complete and densely ordered semigroups G, G' are
isomorphic, and the isomorphism is unique once a, a' in G,
G' are given to be mapped to each other (para 92-93) ,
any such G is the additive group of positive elements of a
field, unique by choosing the multiplicative unit, and the
isomorphism from G onto G' preserves the multiplication (para 110) ,
once we go from semigroups to groups. Of course, Bettazzi
did not use the word 'categorical' for this axiomatization.
(By the way, this means that there is essentially only one
copy of the reals, while there are many ways to 'construct' it.)
As for "Toepell", I am afraid that I never heard of him.
As for the question who influenced Huntigton at Strassburg,
it would be important to know who was his thesis advisor
there. I doubt that he was influenced by Theodor Reye, but
could conceive that he was supervised by Heinrich Weber who
held a chair at Strassburg from 1895 to his death in 1913 .
As witnessed e.g. by his papers on elementary set theory, he
was a man always rerum novarum cupidus.
W.F.