Generally speaking, I believe intrinsic motivation is always stronger and more lasting than extrinsic, and entails fewer risks (e.g. feeling entitled to rewards for routine tasks).
Extrinsic vs. intrinsic motivation is touched upon in Why should children be rewarded for good behaviour? and How do I deal with a child that refuses to do a chore or task?
The problem is that intrinsic motivation can be hard to provide.
We've been taking a similar approach to what you've described, although up until now, our extrinsic incentives have been limited to a couple of m&m candies when my son (now 3 and a half) pees on the potty, and the promise of an aquarium for when he's completely potty trained.
It has been... slow.
My son will happily try the potty most times if you suggest it, and will go through phases where he'll ask to go to the potty once or twice a day (especially if we're at a restaurant), but many times he doesn't actually have to go, and we've only had success with one bowel movement on the toilet (very early on, and likely an accident, as he seemed very surprised).
We discussed it with our pediatrician, and she cited lack of incentive. My son is happy to get the m&m's, but that's just a minor treat. He's excited by the idea of an aquarium, and keeps asking for it, but our pediatrician feels it isn't "real enough" for him, as it is just an abstract concept.
Her suggestion was to take him to pick out the fish and aquarium, and then make a chart with concrete goals to work towards (e.g. "Pee on the potty three days in a row and you get a guppy; go a full day without going in your diaper, and instead going on the potty each and every time, and you earn the aquarium; etc."). Once he's achieved the final goal (full potty training), he gets all the rewards he's earned.
So the professional advice we've gotten has been pretty firmly in favor of providing more extrinsic motivation.
We're currently working on developing the goals and the chart, so we haven't yet started that (although we did take him to pick out the fish and aquarium ornaments he wants).
My perception is that my son's peers who have older siblings tend to go through potty training a lot faster than my son. I'm not sure if that is simply due to the parents already having experience, or the intrinsic motivation of being able to be more like the older siblings, but I suspect that peer pressure from older siblings is one of the more effective intrinsic motivations out there.
I've heard mixed results on some other methods of providing intrinsic motivation, like "potty parties".
