I know this is contentious and will require a lot of discussion, but in my opinion it needs to be corrected. Disambiguating which was the original implementation will help us avoid more confusion with future forks.
IIRC the bitcoin team embraced Bitcoin Core as an overreaction to the manipulation campaign about "control" over Bitcoin protocol, and whether it was clever or just good timing by Gavin to suggest the rename is hard to say.
I was tempted at opening a pull request with the name change everywhere, but think that was silly. If that's something helpful, I'm happy to do it.
There maybe some fear of it being "official", and as we know nothing is "official" in a consensus network like bitcoin, but it is the original client. Nothing stays the same forever, it is possible the consensus of the network looks at a different reference implementation in the future. That's fine, and the name shouldn't matter, but the bar would be higher than just trying a semantic name attack as the recent forks did.
Maybe I'm mistaken, but there seems to be a lot of support for this or if not, we should find out very quickly and i will be the first to put it to rest.
Sorry for the wonderful ESL writing.