> By that definition the abacus is the first computer.
It isn't. The abacus does not compute anything at all unless used as a tool by a human that executes the algorithm. The Antikythera has its algorithm built in, it will compute the same numbers no matter who turns the crank. It's a stored program computer where the program is stored in the proportions of the gears in its gearbox.
Think of an abacus as an aid to manual computation, and a Antikythera as something that actually computes.