In the simplest terms your generator, like most generators in this size range do not themselves generate 240V. Instead they generate 2 different lines at 120 V (labeled L1 & L2) which are out of phase by 180º. It's commonly called split phase
power.
Google it. The 2 circuit breakers shown on your attachment are for these 2 lines of 120 V and are sized @ likely 40 amps to protect the wires that run to your breaker panel. The
funny looking arrow thingy indicates that these 2 breakers have their levers physically connected together, which is required on all split phase lines so that if you overload one line both trip and no
power is available anywhere until you reset the breaker.
These 2 lines of 120 V are brought to your breaker panel by about 8 gauge wires (likely red and black) where about half of the 120 V breakers are connected via a copper bus bar to the red wire and the other half, to the black wire.
Any appliance that requires 240 V AC (your
water heater, clothes dryer, etc.) will have both the red and black connected, which then results in a 240 V supply.
The L0 line is the neutral conductor which is only require for the 120 V consumers.
If indeed you have a 240V supply to some appliance, by definition you must have 2, 120 V lines that are out of phase to produce this 240 V. This would mean both lines must be working correctly. If so, what is the problem?