Is bitcoin.org or bitcoincore.org the one to trust?
It looks like https://bitcoincore.org/ is the one run by the people who run the Github bitcoin/bitcoin project (the actual core developers) since it is the one listed on their Github page: https://github.com/bitcoin/bitcoin
If you want the absolute best source for the client directly from Bitcoin Core devs then I would use that site or their Github page for downloads.
https://bitcoin.org is still, however, a reliable source for information on Bitcoin Core, and maintained by the community. However, it is owned by someone who isn't tied to the Bitcoin Core project itself (as far as I'm aware): https://github.com/cobra-bitcoin