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.
You are correct: "distinguished" doesn't mean that the points are different from each other. A "distinguished point" in an algebraic structure can be thought of as a point that has a label attached to it: in a ring, "I AM THE ADDITIVE IDENTITY!" for point 0, and "I AM THE MULTIPLICATIVE IDENTITY!" for point 1. In the trivial ring, a point can have both these labels attached to it.
set-apart-from; notable; noted; eminent; of particular importance; distinct, distinctive.
To disambiguate the different connotations of "distinguished", we need context.
E.g. In algebra, e.g. in a ring or field, $0$ is distinguished by the fact that it represents the additive identity, and $1$ is distinguished by the fact that it is the multiplicative identity. So $0$ and $1$ are certainly distinguished (set-apart-from other elements, of particular importance, which happen also to be distinct, except for the case of the trivial ring).
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.
If you don't recognize that many expressions are simply abbreviations, and that meta terms are freely used, you will quickly get things wrapped around the axle. The statement that 0 and 1 are distinguished points is an abbreviation for the statement that 0 is a distinguished point and 1 is a distinguished point, and the term 'distinguished' is a meta term, with no object-level mathematical meaning, being used for purely psychological reasons, that is, as a sort of verbal highlighting.