It's definitely not the test webcast which had a completely different set list.
My notes from that time show four webcasts from the original incarnation of Roundhead in Parnell in 2001:
There was the test webcast on 16th January broadcast on nilfun.net which was announced at short notice, followed by the official, advertised webcast on 17th January, sponsored by dotmusic.com. These were Neil's first ventures into webcasting.
It looks like the date you're looking for is 10th April 2001:
Neil performed songs via a direct video link from Roundhead to EMI offices in Brussels, Belgium for contest winners, taking requests. It wasn't broadcast publicly and a Belgian web site www.Cube.be were involved. A few of the songs performed were Throw Your Arms Around Me, Last To Know and Lucy's Song - a Betchadupa song.
My notes seem to show that this private webcast to Brussels then switched over to a publicly aired webcast at some point or was directly followed by a separate one the public could view during which he performed:
Driving Me Mad, Last To Know, Wherever You Are, Ray, Turn And Run and Four Seasons In One Day.
However, this is the only webcast I can't locate an audio recording of, and it's odd that it's not widely mentioned on the forums, so it's possible the whole thing was a private video link to Brussels. The fact that Neil was taking requests on the phone seems to point to those youtube clips coming from the video link webcast to Brussels as the requests on all of the other webcasts were selected from fan e-mails, no phone number was issued/involved. I'm wondering if those youtube clips originated from nulfun.net back in the day, because I've certainly seen them elsewhere.
There is actually one reference to the EMI Brussels webcast promoting One Nil on the Neil Finn forum:
However it's confusing as it was posted on 11th March 2002 - a year after the event. Neil wasn't promoting One Nil at that time. He was on a U.S. promotional tour for One All that started on 10th March 2002 in San Francisco and ended on 22nd March 2002 with a guest appearance along with Ian Hunter at a Jon Brion show at Largo on 22nd March 2002. In between he performed headline shows at Largo and the Coach House in San Juan Capistrano and gave three performances at the 2002 SXSW Festival in Austin, plus did sets on radio at KGSR in Austin and KCRW in Santa Monica.
So it looks like the mention on the forum came a year after the Brussels event occurred.
As you mentioned the final 2001 webcast was on 13th November and featured Shon Sullivan, Lisa Germano, Scott McPherson, Sebastian Steinberg, Dave Dobbyn and Tim Finn.
I'd be interested see if anyone, especially any fans in Belgium can confirm/clarify more details about the nature of the 10th April performance(s)?