Conventional is how most of us normally move. In sports it's athleticism and in many aikido contexts, "not using strength" means reduction, by various means, not avoidance. Even full-on faked no touch throws involve some use of the conventional muscles to make Jedi gestures.
The unconventional usage I have come across means the most obvious muscles are flaccid. I mean hanging, or even dangling like Osensei's pectorals while moving powerfully regardless of the attempts of others to prevent the movement. This is kokyu power (Note: Kokyu does not just mean breath). Some of the IP proponents on this forum seem to have cogent explanations for how it's done.
Hmm. Ok. I'm afraid you can count me as skeptical. I've been told "don't use strength" to move, and when things work really well they seem to have needed no effort, but humans can't move unless muscles contract. It may be a 'conditioned reflex' that we don't have to think about, it may be something that's well trained and can be done in a very "relaxed" manner, but movement can't happen without muscle contracting.
I've tried reading some of the IP info but I get confused because of my (long ago) biomechanics research background. I'm sure there's something in it, but I just don't get it.
I've also discussed with some of my sensei in the past "I'm trying not to use my strength." and have been met (by at least one shihan) with "You got it, use it, as long as your technique is good".
Discussing this sort of stuff with an Iwama-type godan today (he's teaching at a school nearby and I was doing classes on my sport - rowing) - and he said - yeah, right, what were the shihan doing when they were young - swing a sword 1000 times a day, then a jo 1000 (or 5000) times a day, and you're going to develop muscle.