Does the sensor size of the camera (full frame or APS-C) make a big difference when using this article (https://fstoppers.com/lenses-architectural-photography) as a guide to picking the right lens for architectural photography?
e.g. I'm starting out using Canon EOS 10D, with a low budget and trying to decide between a 10-22mm lens and a 17-40mm lens.
Yes the sensor size is important, you're using a 10D??? Generally the camera body is the least important component of your gear, but that body is nearly 13 years old. I would try to get something 4-5 years old max.
For crop sensors, you'll want the 10-22. You shouldn't shoot everything ultra-wide just because you can though. The wider your focal length, the more you distort the space.
The sensor size directly impacts the perspective. APS-C vs Full Frame is 1.6x I think... meaning:
24mm on Full Frame = ~38mm on APS-C
17mm on Full Frame = ~27mm on APS-C
I use a 24mm lens on a full frame sensor for about 90% of my architectural work. If you wanted that perspective you'd probably be best served by the 10-22mm, which is actually ~16-35mm, a great range for architecture. This is not a tilt-shift lens though, so being able to go all the way to 10mm/~16mm will give you some additional flexibility.
Also, the 10-22mm lens only works with ASP-C sensors, meaning if you ever upgrade to a full frame camera you'll need to get a different lens (e.g. the 17-40mm).
A used Canon 5D Mark II is a fantastic full frame camera if you're using a tripod. Given its age it'll struggle in low light situations without a tripod for long exposures.