Spag- yes, that's basically it- as long as one of the parents was born in the UK/naturalised/registered as British themselves.
The confusion comes because of the word "registration". Like, dd, who was born in Italy, is Brit by descent, registering her birth at the consulate wouldn't have changed that. If you are by descent, you are by descent, and cannot pass on your c/ship. However, her children, (if born abroad) will have the right to be "registered" as Brits under certain sections of the Nationality Act.
So, you have 2 kinds of registration going on, registration of a birth abroad (under the Births, Marriages and Deaths act ) which just gives you a bit of paper confirming you've been born (like you might not have noticed!) and registration of nationality (under the Nationality Act) which does of course concern c/ship.
You're right, they should make it clear, I've answered loads of queries on MN over the years from people who think registering a birth automatically gets them a passport or British nationality.