Hey I got it to work!

Recorded my first replay from the old days. Not a GT1 replay, but a GT2 one, so therefore maybe I'll put it in the GT2 section. Oddly, I haven't gotten GT1 to work yet (it just greys the screen if I try). I am trying to document all my old replays on YouTube, you see.
To anyone who's having trouble out there, here is what I did. All of this assumes you've got a capture card already of course, have tried to get something recorded, but it's not working. It also assumes you've got a desktop computer (preferably with windows), a flatscreen TV, and a PC monitor.
Firstly, I did not have to use a PS3 to do this, I was able to work from my PS2 slim. For a capture card, I've got a Hauppauge HD PVR 2, which comes with a pair of
5-way component cables, male on one end, and female on the other. Oddly, I could not get the replay to show on screen with a component cable though, even if I had Y CB/Pb Cr/Pr selected in the System Configuration menu.
So that's the first step. I went into the PS2's System Configuration menu, found Component Video Out, and selected RGB.
Screen Size 16:9
Digital Out (Optical) On
Next step, just connect an ordinary composite cable (this is the yellow, red, and white cable which normally comes with a Playstation) like so: the yellow male end goes to the blue female end. Leave the green and red video connectors unhooked.
Hook up the audio (red and white) to the red and white female ends.
On the back of the capture card, there should be a mini USB port which connects to an ordinary USB via provided cable. This goes into your computer. There is also an HDMI Out port which connects directly to a flatscreen TV. You don't NEED to have a TV connected though, but it helps to have this connected because the HDMI broadcasts a signal in real-time, while the USB signal going (through the capture card) usually has a couple seconds of lag. It helps to be able to see the replay in real-time, or if you're gaming, obviously you want to be able to game in real-time, not with lag, so you don't wind up driving into a wall.
Turn on the game, and on your computer, obviously, turn on the capture card's program.
In the capture card program, make sure Video In is selected to Composite (not Component, or HDMI, or any other settings).
Make sure Audio In is selected to AV In.
Hopefully, you'll be able to see the game in the capture card's program screen.

For the actual video recording, I chose Medium quality, though the Hauppauge goes from Low to Very High. I figure Medium will provide some decent overall stability (no lag), and I don't know if selecting High or Very High will make much difference for such ancient technology anyway.
Wish me luck on getting Gran Turismo 1 to work.

I really want to post some of my very earliest replays from 2003-2004.