In what case is an oral contract legally binding, if ever?

For any case except real estate transactions. And of course, just as with a written contract there are so exclusions: A contract to commit a crime is not binding, for example.

Now I'm not sure if this is the whole picture, but I gather part of the threshold for an oral contract being binding, is where both parties can be said to have 'intended to create legal relations', which is a definition which has been worked over a lot in caselaw and has quite specific meaning.

But there might need to be other standard essential elements of a contract, including 'consideration', and a price (?), and a few other things too, in order for it to be binding. Also it can't be for anything unlawful. Or for real estate.

Any time there is a bargained-for exchange over a matter that does not legally require a written contract. (Real estate contracts have to be in writing.)

As long as the plaintiff can prove there was an agreement, the contract is legally binding.

When there are credible witnesses to the contract, or credible evidence.

If you get into a taxi and the driver delivers you to your address, you must pay for the service. You will be held responsible in a court of law, even though you have no written contract.