Use the right colors

First off, if you're wiring to US standards, you've got the wire colors wrong.

Green and bare copper are both used to identify an equipment grounding conductor. White is typically used as a grounded (neutral) conductor (grey is also acceptable), though can be used as ungrounded (hot) if there are appropriate markings at each termination. From there, basically anything that's not green, bare, white, or gray, can be used as an ungrounded (hot) conductor.

Wiring L14-30 Plug

To properly wire a NEMA L14-30 device, you'll need 4 conductors.

Proper Wiring

In your situation, I'd pull an extra red or black conductor from the panel to the receptacle location and remove either the green or bare conductor. Then you'll connect the original black conductor from the breaker, to X on the receptacle. Connect the new red/black conductor from the other terminal of a double pole 30 ampere breaker, to Y on the receptacle. Next connect the white wire from the neutral bus bar in the panel, to the N terminal of the receptacle. Finally, connect the green/bare conductor from the grounding bus bar, to the G terminal on the receptacle.

Your math is on the money

You are indeed spot on that your heater will draw 9.375A when connected across a 120V supply.

And your 120V wiring is correct as well

You are indeed correct that the two water heater hots should be connected to hot and neutral of the 120V supply, and the supply ground should ground the water heater.

However, your 240V wiring is a miss

You have the wrong plug and socket configuration for your 240V wiring though. While it's not unsafe currently if it was wired directly back to the main panel as your appliance is 240V only and thus needs no neutral, the NEMA 10 was intended for 120/240V apps, not 240V-only apps like yours. The correct plug and socket is a NEMA 6-30, which is "hot-hot-ground" instead of the "hot-hot-neutral" of the NEMA 10 series.

I'd replace the NEMA 10 with a NEMA 6 and then either:

1. use an adapter cord wired as you describe (with 14AWG wire) to plug the NEMA 6-30 into a 5-15 (regular) receptacle. Make sure to label the cord as "FOR WATER HEATER ONLY" so someone doesn't use it on an appliance that will pull 30A at 120V -- a 15A/250V fuse in the adapter assembly would be a wise thing to have, even, as protection against such oopses.

2. change cords when making the generator connection, as your water heater has a terminal box on it that can be used -- a 14AWG power cord with screw leads at one end and a normal 5-15P at the other can be used, wired as you describe. Make sure to wire up the water heater end first, and not to leave the unused cord connected at either end!

How to hardwire this switcheroo (properly!)

If you want to configure this as a hard-wired sort of thing (for instance, if you already have a transfer switch or generator interlocked subpanel installed), it's not impossible (or particularly expensive), it just requires a bit of unconventional (but still Code insofar that putting a 240V water heater on a 120V supply can be considered Code to begin with) wiring.

You'll need for this:

• A Reliance Panel/Link TRB0603D manual transfer switch
• Some /2 with ground cable of the appropriate type and a gauge that matches the existing wiring from the generator inlet
• An Eaton BRWH220 breaker -- you'll probably need to have your supply house order this in for you, but it shouldn't be terribly expensive
• Wire nuts and THHN that match that existing generator cable for a bit of pigtailing
• And inch-pound torque tools and bits for assembly

First off, make sure that existing breaker for the water heater is off, and that your water heater is a conventional electric tank heater with an electromechanical thermostat and no existing connection to the mains neutral (putting 120VAC on a heat pump or hybrid water heater will lead to severe compressor damage, while tankless electric heaters are simply too beastly for any reasonably sized residential generator to have a hope). You'll also need to torque all the lugs you're fitting wires to the manufacturer's specification -- this is a very good reliability measure in addition to being a new requirement (110.14(D)) in the 2017 NEC.

Pick a spot where you can cut the water heater branch circuit and run a cable conveniently to the existing transfer means, then cut into the water heater circuit and install the new transfer switch there. Land the existing hot wires from the breaker on the lugs of the new transfer switch's "utility" breaker, and land the hot wires going out to the water heater on the main lugs inside the transfer switch (they're the two large lugs attached directly to the busbars). Land the ground wires on the ground bar.

Now, run that new length of /2 cable from the existing transfer means to the new transfer switch. At the existing transfer means, splice into one hot leg with a THHN pigtail and a wirenut -- the pigtail routes to the terminal the hot leg in question once landed, and the black wire from the new cable goes into this splice as well. Do likewise with a matching pigtail and the generator's neutral wire, this time hooking the white wire from the new cable into the splice, and land the new ground on a ground bar. The result is a generator-only 120V feed to the new transfer switch.

Now, install the BRWH220 into the new transfer switch, preferably on the same side as the generator breaker. This breaker is special in that it has a pair of isolated line lugs instead of taking its line-in from the busbar stabs, thus allowing it to be hooked to a circuit of different characteristics than what the loadcenter it is in is connected to (say one on a separate meter, which is the originally intended application for this oddity of a breaker). Take one of the outside lugs (either one works, they're both line lugs) and land the black wire from the new cable on it. The white wire from the new cable goes to one lug on the "generator" breaker already installed into the new transfer switch, and the new cable's ground wire goes into the new transfer switch's grounding bar. Last but not least, use another piece of that THHN from earlier to connect the inside lug on the same side of the BRWH220 that you already wired to (that's the matching load lug) to the remaining lug on the "generator" breaker in the new transfer switch.

Button everything back up, turn the water heater's breaker back on, and make sure the water heater runs both off mains and off the generator, with correct voltage (240V for utility power, 120V for generator power) at the terminals of the water heater. Now, when the power goes out, you'll be able to transfer the water heater to generator power and have it run at half-voltage (quarter power) without fussing with cords or the likes.