On which axis do we measure angle of attack and airspeed? The lift value depends on the relative airspeed of the plane. That's what every website says without anymore explainations.

But as we are talking about lift, are we talking about the relative speed ONLY on the axis of the chord line of the airfoil ? Or the relative speed, not related to any particular axis ?

Now about angle of attack (AOA ):

Is the AOA measured on a specific axis ? Or is it the absolue angle between the airspeed Vector and the chord Line ? 
Or with other words : does AOA = 20° when the plane is on a 20° side slip ?? That's seems wrong to me but if we relate to the words of wikipedia for example thats what is being said.
Airspeed is always the total magnitude of the free-stream incident on the wing/airplane. In zero sideslip, angle of attack is exactly the angle between free-stream and body x-axis (could be chord for an isolated wing). In non-zero sideslip, it's nuanced; see below.

By industry standard (Ref. Etkins, Dynamics of Flight; Stevens, Aircraft Control and Simulation), the angle of attack ($\alpha$) and the angle of sideslip ($\beta$) are defined as Euler rotation from the coordinate axis attached to the free-stream such that the speed vector is $\begin{bmatrix}V_a &amp; 0 &amp; 0\end{bmatrix}^T$, where $V_a$ is the airspeed magnitude, to the body frame (where x-axis aligns with the chord):

$$\begin{bmatrix}u_a \\ v_a \\ w_a\end{bmatrix} = \begin{bmatrix}\cos\alpha\cos\beta &amp; -\cos\alpha\sin\beta &amp; -\sin\alpha \\
\sin\beta &amp; \cos\beta &amp; 0 \\
\sin\alpha\cos\beta &amp; -\sin\alpha\sin\beta &amp; \cos\alpha
\end{bmatrix}\begin{bmatrix}V_a \\ 0 \\ 0\end{bmatrix}$$

where $u_a$, $v_a$, $w_a$ are the incident speeds in the body frame.

Simplify, and we have:

$$\alpha = \tan^{-1}\frac{w_a}{u_a}$$

$$\beta = \sin^{-1}\frac{v_a}{V_a}$$



(Image Ref. Etkins, Dynamics of Flight)

Note that when sideslip is small (whereby $\cos{\beta}\approx1$), we can reduce the above to:

$$\alpha \approx \sin^{-1}\frac{w_a}{V_a}$$

Therefore, taking angle of attack as the projected angle between chord and free-stream is pretty darn good.