What happens when an airline carries a passenger with the wrong classification of visa?

First your assumption that the airline staff should know what the symbols mean is a bit unrealistic. There are 180+ countries on this planet, many with different visa categories and different rules for citizens of the other 180 countries. The counter agents check in many flights and many different citizens daily, they can NOT be expected to know every visa variation. Rather they use a database and it says a UK citizen needs a visa AND it seems the traveler had a 30 day visa.

The fact that embassy issued an incorrect visa likely frees the airline from fines for flying your friend there. As for the airline's responsibilty to your friend, your friend presented a 30 day visa and the airline fulfilled their obligation by flying them there.

The entity that bears responsibility is the London Chinese Embassy which issued an incorrect visa. But good luck getting compensation from them.

After your last edit, I now understand that you have a completely different question than the one I addressed originally, namely whether the carrier has any obligations towards the passenger being bounced at the border.

That would depend on the conditions of carriage and local consumer protection laws but that's almost certainly not the case. Passengers are entirely responsible for making sure they can enter their destination and/or transit points and you are not going to get any help from the airline if that's not the case (see also Do airlines have to check the visa status before selling a ticket?)

In that case, there was a small discrepancy with the visa which the passenger could in fact have noticed but it would be exactly the same even if the papers were completely in order. Theoretically, a suspicious attitude or incoherent answers during the landing interview could also lead to entry being denied and nobody (not the airline and certainly not the state who took the decision nor their agent/officer) is going to compensate for the ruined trip.

Airlines check visas, not to help travellers, but because they have some obligations towards the states where they are operating, which is the question I address below.

It's impossible to know without more details and even then it would be mostly guesswork. Generally speaking, airlines can be fined for failing to exercise a certain level of diligence in checking passengers' travel documents and visas but not merely because someone is denied entry for reasons they had no way to suspect.

Beyond that, the exact procedure would depend on local law but airlines can and do go to court to fight such cases, not only to avoid paying the penalty but also to establish precedents and limit future liabilities. For example, here is a recent court case about a slightly different issue that stemmed from a €2850 fine imposed on Air Baltic for carrying a passenger with a visa in an expired passport but without visa in his new passport.

To give you another example, French law (specifically article L625-5 of the CESEDA) provides that airlines and other transportation operators cannot be fined if the documents presented by a passenger had no “manifest irregularities”. Whether the wrong type of visa (as opposed to no visa at all or an expired visa) rises to the level of a manifest irregularity would ultimately be up to the court system but the point is that airlines cannot always be held responsible when someone is refused entry.

Similarly, there is no fine if the person is able to file an asylum application that is not “manifestly unfounded”. Here again the distinction is between an application that is ultimately found to be without merit and one that's obviously ludicrous or fraudulent; airlines are supposed to filter out the latter but not to become judges of the former. I don't know the detail of the law in other countries but I think that all this is pretty typical.

Finally, I am not entirely sure of that but I suspect that an airline could still be forced to bring a passenger back to his or her point of departure, even when it's not found negligent enough to warrant a fine.