Thanks, this explanation and discussion is very helpful. It does confirm how I suspected things to work.

I have trouble understanding the need for the "signing" part of client-side certificates. Currently if I create an account at a website with a username/password, there's no need to get my account signed by a trusted third party. So why not let me create the account with a username/publickey instead? Why does a third party CA need to be involved?

And actually (as I mentioned in another post just now), one thought is to have keypairs supplement passwords rather than replace them. Basically when I move to a new device, I can still log in to the website with a password, and then the site will give me the option to add the device's public key so I can seamlessly log in automatically next time.