FWIW here is what the corrupted image looks like.

Your ATI cards seem later than your nVidia cards, so we could be looking at a 4x/8x AGP card problem vs a 2x/4X AGP card. If the nVidia are falling back to 2x, it's possibly that it's having trouble maintaining voltages for 4x... several boards were kinda sketchy for that in the 2000-2003ish era, claimed 4x support, but if you put in cards that used it, they had weird problems.

That doesn't seem right.
It's clearly doing the math accurately.

To me it looks like the sort of output you would see when an interlaced image is sent to a non-interlaced screen.

Interlacing sends every second line per frame.
Eg 640x480 would send 640x240 every frame with the whole image taking 2 frames.

Are there any settings at all related to that? In the screen or control panel?

Using a vga adapter?

No settings that I can see. And again, this only happens in games (unless they lock the system with a black screen, which some of them do). The desktop appears completely normal.
No adapters. Straight VGA cable from the card to the monitor. The same cards used with the same cable and monitor work correctly in games when used in different motherboards.

Also, while the 9600 is a tiny bit later than the nvidia cards - the 9200/9250 were contemporary with the GF3, and the 9700 with the GF4. All released 2001-2002.

The core design was contemporary with the GF3 as 8500s, they weren't released in that form until the 9700 flagship to be a bottom end range, some of them were minor core shrink versions. Here's the 9000 in mid 2002 getting launched, which means it ain't really around much until Q4 https://www.anandtech.com/show/946 and then the 9100, 9200, 9250 kinda snuck out the following year. GF4 had been out a while by then. Anandtech doesn't even put GF3s on the same chart to test the 9000s. Anyway, they're later cards than your Geforces.

What video modes does the card have in its list, what is the game set to, and what is the monitor set to?

https://drivers.amd.com/relnotes/catalyst_77_ … ease_notes.html

"Connecting an HDTV using the Component Video connector no longer results in a double image being noticed when pressing the ALT+Enter keys. Further details can be found in topic number 737-28473"

Not the same, but maybe related?

The cards have all the usual resolutions, including 800x600x32bpp which is what I set the desktop resolution to. The game is also set to 800x600x32bpp. The monitor is on "auto" which is the only setting it has.
I have already played with different resolutions, including 640x480, 1024x768 both in 256-color mode, 16bpp and 32bpp (for both desktop and for game resolution). The same thing happens in all cases, with all Radeon cards. It does not happen with Nvidia cards, regardless of resolution.