[DutchmenSport]

What you described is perfectly normal. The amount of delay between the time you initially turn on the faucet and the start up of the pump is because there is pressure in the water lines. If that length of time changes, it's probably because of air in the water lines (system) somewhere. Where? The most logical air pocket would be in your water heater. Although it's not much, but the top of the water heater does trap air and this small amount of air works like an accumulator tank.

Sometimes you can get air in the lines, and the more air, the more it works like an accumulator tank. If you turn your pump off, and your water tank is empty and then you travel, and then you fill your tank at your destination, and then turn the pump on, it will have some air in the lines, which is one way you can get air in the system.