A powered damper cannot be simply added to a WH not equipped for that capability. For such capability, there are interconnects required to prevent the WH burner from igniting unless the damper is first verified to be fully open.
While there will be some standby heat loss occuring through the insulated shell into the unconditioned garage space, the larger area of heat loss will be from the exhaust vent tube running through the center of the water tank. With a naturally drafted unit, the largest volume of heat rising and exiting the chimney or 'B' vent will be while the burner is operating, but heat will continue to be transferred to the exhaust vent by the hot water in storage even while the burner is not operating. You are correct in wanting to stop that heat loss, but there are other methods that are in compliance to code.
Many gas appliances no longer utilize natural drafting and so must rely on a forced draft exhaust fan to eliminate products of combustion. Those devices are often vented horizontally through an exterior wall, thereby eliminating the need for a natural draft vertical chimney. Because tank type water heaters have not changed much in technology, modifications were needed so they could be utilized in homes without a chimney. Many WHs are now equipped with an electric forced draft fan so they maybe exhausted horizontally through an exterior wall.
Because the fan will move a larger volume of air than needed to eliminate the exhaust gases, that additional air will dilute the exhaust gases, lowering the temperature of the exhaust, usually permitting small diameter plastic piping to be utilized for exhaust venting.
A power vented WH will require electricity to operate the exhaust fan. Because the WH cannot heat water unless the exhaust fan is verified to be running, the gas valve will be interconnected with pressure and air flow switches which must be satisfied before the gas valve will initiate the sequence to ignite the burner so as to heat water.
Exhausting the WH horizontally will reduce the amount of standby heat loss exiting the exhaust vent while the burner is not operating. Whether the savings will be noticable for your specific application, I can't answer.
One disadvantage to a power vent WH, during power loss, no further water will be heated so once the hot water in storage has been depleted, that's all you get. I anticipate your current naturally drafted WH utilizes a standing pilot light and does not require external power and so will continue to heat water even during an extended period without electrical power.