What is the reason for the acidic character in phenols?
You are confusing the two terms electronegativity/electropositivity and higher/lower electron density.

The source you read states that the $\ce{sp^2}$ hybridised carbon in phenol is more electronegative than the $\ce{sp^3}$ hybridised one in (say) ethanol. This means, that this carbon has a greater desire to draw electrons towards it.

Therefore, even though any carbon is still more electropositive than oxygen, the phenolic oxygen will be able to draw the electrons towards it less effectively and therefore has a lower electron density than in ethanol.

But the oxygen atom in itself hasn’t changed; specifically, it is still hybridised the same way (assuming that one can speak of hybridisation for oxygens). Thus, oxygen’s electronegativity is still the same.

The combination of still having the same electronegativity but having less electron density means that any other electrons will be drawn towards it more strongly — specifically, this concerns the electrons in the $\ce{O-H}$ bond.

In a nutshell: The oxygen’s electronegativity didn’t change (it sounds weird to talk of electropositivity for oxygen if there is no fluorine around) it only has less electron density on one side due to more pulling. It combats that by pulling more from the other side.

This can be called an inductive effect.