Is the welder's plug hot-hot-ground or hot-hot-neutral? If it's an actual NEMA10-30P there is no separate ground, but I doubt that is the case with the welder. It's not a NEMA6-30P by chance? Is the 3rd prong round or a crow's foot ('L' shaped)? NEMA10-30 was dropped from the NEC after 1996 but is still used on appliances. So no new work can be done with it but they still build stoves and dryers that use it. The difference is the third prong being a shared neutral & ground means you can supply 125V circuits where a 250V-only NEMA6-30 cannot.
I think from your descriptions that you'll want to use the two hots and the ground terminal on the 4-prong but don't assume that is the case. You'll open GFCIs potentially if the 3rd prong is supposed to be neutral and not ground. Not to mention the shock risk of putting current on the ground unintentionally.
FWIW, you should not use a white wire as a hot leg, but in an extension no one will really know (I'd personally tag it 'cause I'd forget why I wired it wrong). Black, red and blue are hot colors, white is neutral and green or bare is ground. The odd ball exception to this is NEMA10-30, but that's really more of a missing ground because the white is carrying current like you'd expect.
"Silent enim leges inter arma." -- Cicero