Hello Fred,
My money is on Ash's answer. But perhaps he hasn't explained why its happening.
In essence you have a pressure switched controlled system that senses the pressure in the cold water supply pipes. The pressure switch opens its contacts when the pressure rises above a threshold. (break on rise) The pressure in the system is locked in by a non-return valve in the pressure switch.
If the pressure drops below a second lower threshold (Make on fall) the pump turn on again.
The two thresholds are not quiet the same, so there is a small amount of hysteresis in the system.
Because the cold pipes have a relatively small volume, and not a great deal of elasticity, it only need the cold tap to be turned on for a fraction of a second for the pressure to drop and the pump to turn on.
The hot system takes a feed from the cold, but it too has a non-return valve. The other major differences are the volume of the hot system, is much bigger, and it is designed to have an air cap in the heater tank. When the tank is initially filled (when you get on site) the water level rises in the tank until it reached the hot out let level. At this point you will get water and air emerging from the tap, but it soon settles down to a constant flow. At this point you close your hot tap, but the pump continues to run. The water level in the tank is still rising, but it is compressing the air cap and the pressure is rising. Eventually the pressure rises enough to cause the cold pressure switch to trip.
Both hot and cold systems are now at the same pressure, but the hot system has the compressed air cap. If the heater is turned on, the water expands and air cap gets hot, which further increases the pressure in the hot system.
If you now turn on the cold tap, the pump will turn on as before, because the hot system does not back flow into the cold because of the heaters non-return valve. But if turn on the hot tap, the pressure in the heater tank will force the water out, and because of the air cap, the pressure drop is much slower in the hot system. Eventually after a few seconds the air cap will have expands fully as the water is pushed out. The pressure now begins to fall below the cold water (Make on fall) pressure and cold water will again begin to flow into the tank to displace the hot water.
For the first few time of using hot water you may get some spluttering of air in the hot water but this soon diminishes as excess air is expelled from the system.
One major solution to this is to add an expansion reservoir (surge damper) to the cold water system that compresses air in a similar way to the hot system. This can reduce the surge effects.