I like the perspective that the set of states is precisely the set of positive trace class operators $M$ of trace one. A state is called pure if $M=pr^{\perp}_{U}$ is the orthogonal projection onto a one-dimensional subspace $U$.
So, every non-zero vector $\psi$ defines a pure state. Since the orthogonal projection onto $\mathbb{C}\psi$ is the same operator as the orthogonal projection onto $\mathbb{C}x\psi$, for every $x\in \mathbb{C}^\times$, there is no confusion about equiavlence classes.
Concerning the superposition of states, one shows:
The set of states is not a vector space, rather it is a convex space. This explains (a) that in a linear combination of vectors only the ratio of the coefficients counts, and it explains (b) that convex combinations $$ aM_1 + (1-a)M_2 $$ of states is the only allowed operation on the space of states.
Every state is a convex combination of pure states.
A state is pure if and only if can be written as a convex combination of other states only in a trivial way.
This point of view is described in the book
- L. A. Takhtajan, "Quantum mechanics for mathematicians", Graduate Studies in Mathematics Vol. 95, AMS, 2008