Inverter is running off of battery
(12 VDC) to provide the 110 VAC. Pretty finite supply of power. Fine for low consumption, say charging your camera or laptop battery, not going to last very long running one of those cube book shelf electric heaters or a 300 watt shop light
Converter typically is hooked in so when you plug into shore power 12 volt power for things like roof vent fan, water pump, 12 volt lights
comes from converter not battery. Converter will also charge the battery so when you unplug from shore power your battery is charged up.
Water pump is (almost certainly) 12 volt and running off of the battery.
If you have a converter that is great, I don't and when I priced them out I figured solar
and a small battery charger was cheaper and better value for my needs.