<I'm not sure, but if the object isn't really in an Earth orbit, it's more in a Solar orbit, isn't it?

It depends how you want to think about it.

For example, in L1 an object orbits a point between Earth and sun perpendicular to Earths orbital plane. It's going around the sun, clearly but the path it would trace out would be an unbalanced sinusoid or a cycloid, not a line.

They're both in orbit around a ray through the center of the Earth AND move about the sun at about the same angular position and velocity as the Earth.

This would seem to be one of those, "is Pluto a planet?" questions. Oui? ;)