I'll let somebody else come along and tweak my explanation, but the 4-pin is what most utility trailers use. It just sends a signal for blinkers, running lights, and brake lights basically. The 7-pin connection adds 12-volt power and braking.
Anybody who has a 7-pin plug on their truck and has rented a trailer from U-Haul now owns the 7 to 4 pin adapter. Thankfully I've saved myself $30 several times by always having my adapter in the console.
On previous TVs, I had a hitch receiver installed by a dealer who put the 7-pin plug by default. On another one, I negotiated for the dealer to add a hitch receiver and they farmed it out to U-Haul. U-Haul put a 4-pin plug on and I had to fight them to get the 7-pin added. Then I had to fight them to actually run the extra wire for the 12-volt.
One time I sold a popup to a family from Australia who was towing with an Isuzu Rodeo. U-Haul had installed their hitch. They had a 4-pin plug. I wanted to close the deal so I towed it to their house and showed them what adapter to buy.
Looks like you're just ready to use either connection. If you have a converter/inverter and a battery, the 4-pin won't charge the battery from the truck. Also, if you have a 3-way fridge, the 4-pin won't run the fridge on 12v through the truck.
DW, DD (2007), DS (2008), DD (2010), DD (2011), and me
2010 Chevrolet Express 3500 LT
6.0 liter V8 SWB 12-passenger van
2008 Fleetwood Utah Popup Camper