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To answer your direct question. Velocities are instantaneous and therefore there is no such thing as Euler notation vs fixed frame notation. Also note that quaternions represent orientations and are not appropriate for velocities.

You can convert between all the different conventions, Euler*, Fixed axis etc. You will also find that EulerZYX(YPR) and FixedXYZ(RPY) is conveniently the same three components.

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