Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set.
At the same time, it defines convex combination as
In convex geometry, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.
I might be very picky, or mistaken, but I'm led to believe that if $|\psi_i\rangle$ are pure states spanning whichever Hilbert space we are dealing with, then (say) $|\psi_1\rangle$ is a pure state and also a convex combination $$ |\psi_1\rangle\langle\psi_1|=\sum_ia_i|\psi_i\rangle\langle\psi_i|, $$ with $a_1=1$ and $a_i=0$ if $i\neq1$.
Am I misinterpreting something? Is Wikipedia unclear/wrong?