Yes, moving charges can repel and attract each other, but that is not only due to their electrostatic fields. The force that one moving charge (let's say A) applies on the other (let's say B) is the net resultant of the force due to the magnetic and electric fields produced by A. To define solely electric charges, we need to use their most basic property of producing an electric field (and thus exerting a force on another charged particle). So we use the term "electrostatic" which ensures that only the static electric field (which is by the virtue of an electric charge) is taken into account.
I would say this is not a conceptual physics doubt but rather a question on how a definition is framed (the semantics of defining something).