The word "distinguished" in algebra When we define something like a ring, we often say that the elements $0$ and $1$ are "distinguished elements". What does this mean? It obviously doesn't mean they are distinct.

In my experience, in algebra, "distinguished elements" never implies that the elements are distinct. One usage is to denote named constants (a.k.a. nullary operations) in algebraic structures, such as constants $0$ and $1$ in rings - the additive and multiplicative identity elements, respectively. Similarly one may speak of structures with distinguished maps, e.g. groups with distinguished operators, vector spaces with distinguished linear maps e.g. (when deriving normal forms for matrices), and rings with a distinguished automorphism (e.g. in difference algebra, which studies solutions of difference equations = recurrences).
A quick search of Google Books confirms this usage, e.g. see definitions of rings and fields in Cohn, Skew Fields p. 3; Enderton, Set Theory, p. 122; Jacobson, Basic Algebra, p. 86;  Lidl and Niederreiter, Finite Fields, p.12; Rowen, Ring Theory p. 2, etc.
