I will explain this, but if you don't understand it get an electrician out there to do it. Electricians are cheaper than dead people.
No matter if you have a generator, inverter or grid power, you should have a Main service panel in your house that the power source feeds. There will be a ground rod driven at this, what is called the Service Entrance. This is your earth ground. The Service Entrance is the term given to the point where all the power is distributed in circuits protected by a main breaker for the power source, and circuit breakers for each circuit.
Inside that service panel is a provision to install a green bonding screw that bonds the system neutral to the earth ground. That green screw should be installed in there and this is your one and only neutral ground bond allowed in the entire system.
Now, no matter if you install a generator, hook up grid power, or an inverter, you only run hot to hot, and neutral to neutral, and ground to ground. You DO NOT make another bond between neutral and ground at the generator frame or anywhere else.
The generator you have, when wired for 120 volt, will supply 60 volts from neutral to ground, 60 volts from hot to ground, and 120 volts from hot to neutral. This is because the sine waves are technically split phase 120 volt power being you wired the generator windings in parallel. It has a floating neutral. This type of generator can NOT be wired in to a system with a bonded neutral. Period.
If you rewire the generator for split phase, now you have a good unit with a generator neutral. You have 120 volts from either leg to neutral, and 240 volts from leg to leg.
With the generator wired for 240 split phase you can wire it to your service going hot to hot, neutral to neutral, and ground to ground. However, you will have one hot leg from the generator that is not being used. This means you cut the gen output in half and you will only use half the generator windings. If you want to use the full output of the generator, then you will have to install a transformer that converts the split phase output of the 240 gen down to 120 volt.
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Chris