John, I think you’re confused, or maybe I am. An amp is a measure of rate of current flow. An amp-hour is not a measure of rate, it's a measure of capacity (in this context).
Understand I’m an ME, not an EE, but here’s what I think I remember from EE201. If there is an EE hanging around here, and I’m wrong, please straighten me out.
An ampere, or amp, is a rate of electrical flow, defined as one coulomb (sp?) per second. A coulomb is a defined quantity of electrons (a huge number, IIRC).
An amp-hour is a quantity of electrical charge. One ampere of electrical current flowing for one hour.
When the unit amp-hour is applied to batteries, it refers to the capacity of a battery to supply one amp of current for some number of hours, defined by the point at which the voltage (AKA electrical pressure) drops below some level (10 volts, maybe, for an automotive battery).
To restate the classical water analogy, think of a battery like a water tank. The tank has some capacity, maybe expressed in gallons. If you open a valve at the bottom of the tank, water will come out at some pressure (maybe expressed in pounds per in^2) and flow rate (maybe expressed in gallons per minute). A battery has some capacity, too. It is generally expressed in amp-hours. How many hours will a battery discharge one amp of current (gallons per minute) before the voltage (pressure, psi) drops to an unacceptable level.
The water tank doesn’t care at what rate you pump water out of it; slow or fast, when it’s gone it’s gone. If a battery will supply 6 amp-hours of electrical charge, it isn’t really relevant (within reason) at what rate you discharge it.