reference frames - Euler angles derivation

I have been trying to grasp the idea of Euler angles for a while. Can anyone point out if my understanding is correct or not.

Situation: We have 3 axes known as principal axes of inertia which define rotation axes of a body, the body is rotationally symmetric about each of these axes, and If we start from a Cartesian $i,j,k$ system, we need to orientate such that they are in position or match the principal axes,each of the orientation the angles rotates defines a unique change in orientation i.e. rotation, so we can construct any rotation by combining these Euler angles vectors?

I don't think I fully understand how the Eulers angles were defined, my intuition to these derivation should be defined like this, 1. rotate about $z$ axis 2. rotate about $y$ axis, 3 rotate about $x$ axis. But I can't convince myself why I'm wrong. I hope any experts can point me out.