On which axis do we measure angle of attack and airspeed?
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.