Loaded voltage is lower than resting voltage. You are not sening battery 'drain' at the initial drop in voltage, but only after that while the load is running. Once the load is removed, battery voltage will bounce back. Not back to where it was, but to where it was minus the drain while the load was running.
With the thin wires to the 12v plug in, you will get more voltage drop under load, so you want to connect the inverter to the battery itself with fat wire for best results.
If the batteries are "weak" (old eg) they will show proper voltage resting but loaded voltage will drop way more than it would when the batts are "strong". You might be seeing some of that in there too.
