Official tournaments from the publisher will be allowed by BGA.com assuming the publisher follows all the international legal requirements correctly, which is actually quite a hassle.
For interpreting the reply I got:
Tournaments held on BGA must not contain any prizes/awards of any sorts, and there should be no mentions about any kind of prizes on BGA.
If you're doing things on your own platform, we take no responsibility, so it will be fine.
The first sentence says we cannot host a tournament with any type of prize and says we cannot ever mention prizes on BGA.com. This would very clearly refer to hosting a tournament and listing a prize in the description or discussion, any kind of publicity regarding a prize.
However, the second sentence is interesting. "Doing things on your own platform" seems to refer to handing all communication regarding the tournament and prizes on a separate website. "Doing things" seems to refer to communication and not toward operating an online board game hosting website. I don't think they're saying "make/use a separate online board game to play the games", otherwise the first sentence would have been sufficient, no need for the second sentence.
Saying "we take no responsibility, so it will be fine" seems to be saying that they would not be legally liable for a tournament with communication on a separate website because they would have no way of knowing about it or being found liable for any issues that arise.