Although the high-frequency switching type doesn't have the surge capacity of the transformer based, they do have some definite advantages. They are much lighter, usually quite a bit smaller, and (especially in the lower power ranges) they are much cheaper. However, if you are going to run something like a submersible well pump, you will need either very high surge capacity or you will need to oversize the inverter above its typical usage, so that even at maximum surge the inverter will not exceed its surge rating.
Typical or usual power is what the inverter has to supply on a steady basis. This is the continuous rating. This is usually much lower than the surge. For example, this would be what a refrigerator pulls after the first few seconds it takes for the motor to start up, or what it takes to run the microwave - or what all loads combined will total up to.
Average power would usually be much less than typical or surge and is not usually a factor in choosing an inverter. If you run a pump for 20 minutes and a small TV for 20 minutes during a one hour period, the average might be only 300 watts, even though the pump requires 2000. Average power is only useful in estimating battery capacity needed. Inverters must be sized for the maximum peak load, and for the typical continuous load.