So what's the difference in performance of cleaning solvents that are branded as copper solvents and general cleaning solvents that claim to remove copper fouling in addition to powder residue? Do the copper solvents work better at removing copper? Are the general solvents more effective at removing powder residue?

Some like the old Hoppe 9 clean carbon decently and will with enough work get the copper too. I would include Butch's,TM and Hoppe's Elite copper cleaner in that category too. If you have time and let them work they will eventually get the copper out.
So to answer I would say yes copper solvents work faster for sure. The combo stuff will get there eventually.

Myself I prefer just to put one of the foaming cleaners in let it soak a while then see what the patch looks like and reapply until I am getting clean patches. If the original foaming shows lots of blue indicating copper I may use KG12 copper remover.
I borrowed a Hawkeye bore scope a few years back to see what worked and what didn't just to answer the question you asked.
If you know you have a serious copper issue JB bore paste is your friend and will make short work of it.

There is a product available called Dyna-Tek Bore Coat. When used as directed, it will significantly reduce metal fouling and make cleaning much easier. It ain't cheap though. I have a couple of kits waiting for the off season and I intend to treat one or two rifles to see how I like it, and if I do, I will treat the rest of them. I have a couple of rifles that metal foul heavily.

The foaming cleaners do a great job of getting copper fouling out. I foam it and then put it barrel-down over a paper towel.

I've switched over to Wipe Out Foam and Wipe Out Tactical Advantage. Leave it in overnight and wipe out with a patch. Really good at removing copper, powder and carbon without using a brush. It has rust inhibitors so it's safe on the barrel.

Another vote for Wipe Out / Patch Out. They are game changers.

Typically your copper solvents are ammonia based I believe. Sweets, Butch's, etc. I don't get that ammonia smell with Wipeout, but it seems to work pretty well. I typically use Hoppes #9 for my general cleaning, probably more for nostalgia, and it's always worked fine for powder fouling. Hoppes is mild however, and really won't do much for copper. The others mentioned do a good job for the rare occasion I feel the need to "deep clean" or have a rifle that has excessive copper fouling. As mentioned, you'll know your getting copper when your patches come out blue. Be aware of false positives if you're also using a copper brush, and make sure to neutralize them after you're done cleaning. Copper solvents will eat up your brushes if you don't.

Harbor Freight sells some foaming thread-cutting oil, sulfur-based, intended for use with a pipe-threading machine. I accidentally discovered that the stuff does an outstanding job on carbon, better than anything else that I have encountered.

I run a patch wet with the cutting oil down the bore, let it sit a few minutes, then run a brush through two, or maybe three times. After that, a dry patch will come out pitch black, loaded up with carbon that has been knocked loose.

I understand brass rod tips will also give false positives.

I've started using Wipe-out foam in the last year or so and really like it. I've also found that if I run a few patches soaked with Kroil down the tube beforehand, it really cuts through the powder fouling and let's Wipe-out do it's job even better.