Which sites exactly? I’ve never seen it actually happen and I’d like to experiment.

Travel sites are worst about it but I've seen it on Amazon which claimed it was a mistake. Smaller stores don't typically do it, but with an iPhone and an Android, one on VPN sometimes you can catch the site being funny about pricing.

Are any of the websites actually the official website of the hotel/airline/car rental?

I’ve never even seen a lower price advertised on a third party website compared to the official website of the vendor.

I've only seen this on third party sites. Never in direct sites. I've definitely seen cheaper prices on third party rather than direct sites on more than one occasion.