So in many sources I have read that

A pure state contains only one element, since the only entry on the 
density matrix will be 1.

But what about superpositions?

I mean a state like

$$\mid \psi\rangle = \alpha\mid0\rangle+\beta\mid1\rangle$$

for example is a pure state but it's corresponding density matrix can have 4 non-zero elements.

Why is it that these definitions seem to ignore superpositions?