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Commonly we use geodetic datum to describe the geoid, a distorted ellipsoid model of the Earth, but is it correct to say Earth-Centered Earth-Fixed (ECEF) datum?

Because we are applying ECEF to a sphere model of the Earth. Datum is the applied model, such as sphere, ellipsoid, or geoid.

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Within the geodetic community, a geoid is a equipotential surface that roughly matches mean sea level if there was no topography.

You might want to take a look at this IERS site on the difference between ITRS and ITRF. ITRS, International Terrestrial Reference System, is the theoretical world system, which is then realized using a set of control points into an International Coordinate Reference Frame. They are always in ECEF (3D Cartesian), and in fact only acquire a particular ellipsoid like GRS80 when you need to convert the XYZ coordinates into latitude-longitude-ellipsoidal height values.

So back to your question. I would probably not use datum nor geodetic datum, but instead reference frame.

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