Angle of attack increases lift for two reasons, Camber and increased flow deflection, What reason does the lift coefficient use? Both?
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The lift equation is:
$$ L = C_L \frac{\rho \times v^2}{2} A , $$
where $C_L$ is the lift coefficient, $\rho$ the density of the air, $v$ the relative velocity and $A$ the wing area. The idea behind the lift coefficient is to combine all aerodynamic effects related to the shape of the aircraft into one dimensionless number, which can be experimentally derived as a function of the angle of attack:
(example lift curve from Wikimedia)
When extending the flaps, two things can happen:
The shape of the wing changes, i.e. wing camber is increased. This will modify the lift curve:
(image taken from this answer)The total area may increase (e.g. for Fowler flaps, see also this question: Does deploying flap change the aspect ratio of the wing?).
So the $C_L$ is taking the change in camber into account, but not the change in area, which will also contribute to increased lift.
