Social democracy and the labour movement is socialist at heart and in origins and it's values and end-goals reflect that. However since the neo-liberal wave of the 80's the party establishments has drifted heavily right-wards, trading massive, fundamental parts of socialist ideology for neo-liberal brainfarts. In Britain, this came to a head with Tony Blair and his "New Labour" which basically threw socialism under the bus and went straight liberal.
Now, Corbyn is more of an actual social democrat. Somebody who still holds to more of the pre-neo-liberal Labour ideology. He came into power because people were getting tired of the "New Labour" nonsense, but because the Labour establishment is fundamentally liberlised by now, has spent basically all his time having to fight his own party members since he has the support of Labour members but not it's leadership (or at least did, I don't know where he currently stands).
So to answer your question of "when Brits say Corbyn is socialist and disliked even on the left for it, do they mean the American definition of 'thing I don't like and don't want to have a productive conversation about it' or do they mean he is true full blown hardcore dictionary definition fitting socialist/communist?": Neither, because that's still a thoroughly Americanised view of things. It depends on if your "true full blown hard-core dictionary definition" includes social democracy and the labour movement as Socialist (which it should, or it's wrong).