They're all called taxis for the most part.
'Black cab' specifically refers to the famous London cars that have a particular shape.
I agree. Black cabs are hailed though, that's the point. You don't call a black cab, that's not how they work. That's what private hire (e.g. Uber) is for.
WRT buses: almost everywhere in the UK is request stop, or at least I wouldn't rely on just expecting the driver to guess.
There might be ten different buses stopping at your stop for example, so it's inefficient to just stop anyway, unless someone is already getting off.