What can cause short on-off bursts of the pump when water is drawn inside the house but normal operation when using an outside hose?

In most cases the pipes that feed exterior spigots are lower than those in your home interior faucets. Air can get into the lines and when it does it rises to the upper plumbing fixtures. The air causes a slight break in water delivery and results in a blast. If it is frequent enough to be a bother perhaps installing air traps in your line is the best solution.