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On all subways I've seen, each car has a red emergency cord. You pull it and the train stops.

Why would you ever want to pull it? Obviously it's to stop the train, but why would you want to? If you think about it, there's really no good reason to pull it. Every subway car has emergency information, and I'm paraphrasing:

In the event of Fire Notify the crew member, exit to the next car. But don't pull the emergency cord, and don't exit the train, the tracks are electrified.
Suspicious or criminal activity Notify a crew member, who will in turn notify the police. (But don't pull that cord, as you will be trapped in that car in the tunnel with no exit. Not a good idea if you're being mugged.)
Medical emergency Notify a crew member, of course, and if you're qualified, inform them. Again, at all costs, don't pull the cord, as you'll be trapped in a tunnel with no easy exit instead of pulling up to a station platform with a waiting ambulance.

I wondered, what exactly is the purpose of the emergency cord if you can't pull it for any emergencies?

I finally found the answer. According to the NYC subway website:

Using the Emergency Cord
Use the emergency cord only to prevent an accident or injury. For example, if someone gets caught between closing subway car doors and is being dragged, pull the cord. But if your train is between stations and someone aboard becomes ill, do not pull the emergency cord. The train will stop, preventing medical professionals from reaching the sick passenger. A sick person is better off if the train goes to the nearest station where police and medical services will be waiting or can be quickly summoned, without interruption.
Emphasis mine.

Albert Herring says re Subway emergency cord: The emergency handle on London Underground trains only stops the train if it is all or partly in a station. for those very reasons.