The "Blue wire" on the Battery Tender instructions (labeled W1) is added to balance the load when charging parallel connected batteries (positives to positives and negatives to negatives)...but not needed for low current chargers
This wire diagram in the instructions is just a way for Battery Tender to ensure you connect the charger to the correct negative lead to ensure proper load balance.
I believe you can leave the BatteryTender connected on the batteries while running the engine/generator - but it is usually a good idea to remove the BatteryTender from 110AC before starting the engine to protect it from the high amp charge that is possible from the engine.
I do not risk my Battery Tender and remove both ends (unplug from the 110 and unplug 12VDC w/ the quick release plug on the 12VDC wires) before starting the engine/generator.
Kim and Steve, Mustang LCDR (Ret), '07 Damon Outlaw #1193
I have seen gross intolerance shown in support of tolerance, Samuel Coleridge
WE LOVE OUR OUTLAW RV