I have an old 5KW generator with an L6-20 and a 5-20R duplex receptacle. From the generator itself there are two black and two white wires going to the distribution panel. Nominal voltage is 120 between black and white, and 240 between black and black. Inside the panel, the two black wires go first to the X and Y lugs on the L6-20 (the ground lug is not connected--that seems odd), and then each black to one of the two hot terminals of the 5-20R. The white wires go to the neutral terminals, and then to the ground terminal. So far so good.

I want to replace the L6-20 with an L14-20, which is now standard for use with things like transfer switches. So my question is, should I

1. simply run the two hot wires to the X and Y lugs on the L14-20, the neutral wires to W, and leave the ground lug unconnected (essentially, what I have now),

2. connect the neutrals to both W and ground, or

3. run a separate ground from the distribution panel to the ground lug on the underside of the generator frame?