I'm completely confused about the difference between geocentric equatorial coordinate systems, ECI, and ECEF. David Vallado, in his book 'Fundamentals of Astrodynamics and Applications,' mentions that the ECI frame is commonly confused with the geocentric coordinate system. In the Body-Fixed Coordinate System section, he also states that he will use ECEF as a synonym for such system. Later in another section, he treats both as geocentric coordinate systems. However, in Curtis Howard's book 'Orbital Mechanics for Engineering Students,' the difference is never mentioned, and ECI and ECEF are treated as if they belong to the geocentric coordinate system.
So the question is, are ECI and ECEF frames and not coordinate systems, and do they belong to the geocentric equatorial systems since both have their origins at the center of the Earth? If I want to explain, would it be like this:
Under the geocentric equatorial coordinate systems, the ECI and ECEF frames can be found?