I have seen that miles can be used for a footsoldier, but does eques or equester refer to a knight on horseback? Are there ant Classical examples of this? Are there any dictionaries that specifically define these terms as a horseman or rider? I'm sure there are plenty of Medieval and Renaissance words, but any from before?
Yes, eques absolutely can mean knight:
II. In partic. A. In milit. lang., a horse-soldier, trooper; opp. pedes, a foot-soldier, Caes. B. G. 1, 15, 3 (twice); 1, 18 fin.; 1, 23, 2 et saep.; “opp. pedites,” id. ib. 1, 48, 5; 2, 24, 1; 4, 33, 3 et saep.; “opp. viri or homines, for pedites,” Liv. 21, 27; 9, 19: equites singulares Augusti, v. singularis.— 2. Meton. or collect., horse-soldiers, cavalry: “plurimum in Aetolis equitibus praesidii fuit: is longe tum optimus eques in Graecia erat,” Liv. 33, 7 fin.; 2, 20; 8, 38; Suet. Galb. 12; Flor. 2, 6, 13; Tac. A. 3, 46; 12, 29; id. H. 2, 89.—
Equites as a social class comes after the meaning of eques as a knight/horseman.
The Lewis and Short entry for eques gives several classical use examples in the sense of "rider" or "horseman". See Caesar's De Bello Gallico 1.15, for example. A foot soldier is pedes (gen. peditis), whereas miles means a soldier in general.
Eques also means an individual from the class of knights. In this context it refers more to a position in the society than fighting on horseback.