So if I understand this, this is how it would work.

At any time you'd have two passwords: one regular, which you use every day; and one for one-time-use only, which you keep around in case of need.

When sitting at an untrusted computer, you use your one-time-use password. This proves your identity, but also immediately expires your one-time-use password. Next time you want to generate a usable one-time-password you'll have to login with your regular password again.

Is that it?

More like the OTP token would be on my smartphone or some other device, so I can generate passwords as needed that are guaranteed to be unique.

But they don't require my normal password to be entered at all. I'd accept punching in my real password or a code on my phone to generate the smart token, since the time required to brute force something like that would be give me plenty of time to revoke it's authorization.

Ah sure, as long as both the client and the server are in agreement of what the next token should be then everything is fine. I didn't think this through.

That would be much more useful. My password is strong enough, but I would never use an computer other than my own to log into Gmail or any other important account. One-time passwords would be a good thing to have.

Maybe possible with 2-factor-auth, but it would still require me to input my password on an untrusted device. No.

Two-factor authentication might partially be the answer to that. You still provide your own password, except that a second password is generated when you request to be logged in. At least that how I think it works.