I assume that your 2001 is the same as a 2000.
The Factory Service Manual for model year 2000 shows that the low-voltage (coil) side of the fog lamp relay is in series with one of the inputs to the Daytime Running Light module. Because of this I wouldn't cut any lugs off the relay or run without the relay.
The switched side of the relay—the two large lugs—looks as though it performs the same function as in the non-Daytime Running Light trucks.
The fog lamp relay is a normally closed relay. It is energized when high beams are selected, interrupting the circuit that powers the fog lights. You can keep the fog light circuit energized with the jumper, but this would involve pulling the relay and interrupting the low-voltage side of the relay/ Daytime Running Light module circuit.
It seems to me that you could have fogs-on-with-high-beams by pulling the fog light relay, soldering a jumper to the two large lugs CLOSE to the body of the relay, and reinstalling the relay. Some TDR members have done a stealth modification by removing the relay cover, jumping the two large lugs internally, and then replacing the relay.