The way you tell is to read the DC VOLTAGE across the battery terminals using a multimeter...
When the SHORE POWER/GENERATOR is disconnected your batteries should read 12.6-7VDC if fully charged.
When you plug in to shore power/Generator your battery terminals DC VOLTAGE should jump up to the 13.6VDC telling you the batteries are being charged.
When you turn off shore power completely the battery terminals should drop back to the 12.6-7VDC showing they are fully charged. Then start your truck engine and the DC VOLTAGE on the battery terminal should go back up to the 13.6 VDC level depending what your truck alternator is putting out at that time...
You have already done this part with shore power/generator connected but you probably need to unplug the shore power/generator completely then start your truck and see if that switches to 13.6VDC coming from your truck alternator system.
Roy Ken