The content of the Maxwell equations are named after Maxwell because he derived their wave character, and theoretically predicted the induction of magnetic fields by changing electric fields (the Maxwell term). This is the reason they are named for Maxwell--- he took the known field relations from empirical laws and turned them into a comprehensive theory.
The motion of point charges was not so much of a concern at that time. I would assume that the discovery of the electron added an impetus to figuring out how points move. Bulk material properties are more interesting to physicists in Maxwell's time, and I wouldn't be surprised if bulk forces appear in Maxwell which reproduce the Lorentz force when you take the limit of a small charged sphere. I don't know, but it doesn't matter at all.
Any other system of equations is presumably equivalent in physical content to the Maxwell equations, predicting the same set of electric and magnetic fields in response to the same currents. If Hertz's theory predicts the same as Maxwell's, it is equivalent, and if it doesn't then it is a new theory, and probably a wrong one.
The question of which particular shape of symbols to express a physical law is a very inane one. It doesn't matter so long as the predictions for experimental quantities are the same. This prediction can be written as Maxwell's equations, as Hertz's equations, as differential forms, or as a C-code to give the fields from the sources. It really doesn't make any difference, it's still Maxwell's theory.