"Do I have to use the same camera I filmed with to dump the footage into
my computer?"
Free,
The answer to that is; 'not necessarily'. Used to be that tapes shot on any particular format would work with any brand of camera using the same format. Now, that isn't always the case. Since you've got two Canon rigs, that might be doable. Since it's just data you're capturing via firewire the issue of 'quality' or 'generation loss' isn't an issue.
When you start crossing up brands like Canon, JVC, Sony and so on that's when you run into problems. Concerning SD shot from other cam's playing in your JVC, you'll just have to test it. Years ago I had a guy shoot a doc for me in SD DV. He used a Sony HDV cam for the shoot. I couldn't play the tapes in the little Canon DV rig we were using as a deck so I thought he shot it in HDV.
Years later, I get a Sony Clamshell to play the tapes on, lo and behold the guy did exactly what I told him to but due to Sony's proprietary BS, the tapes wouldn't play on the canon. Weird thing is, a year later after that job I shot a film using a JVC pro cam at DV and the tapes played just fine in that same little canon rig.
So you'll ultimately have to test your cam's to see which one's will 'play nice together.'