>>> it is illegal to drink in most public places
No it isn't!
In general it's perfectly legal to consume alcohol on the street, in parks or on beaches, etc. Even when there's a 'controlled space' order in place (which only applies to a very small number of places) the consumption of alcohol itself isn't illegal. (It only becomes an offence if you continue drinking alcohol in such an area after a police officer has required you to stop. So, for example, the police can insist that a group of drunken yobs in a 'controlled space' cease drinking their cans of strong lager, while at the same time permitting a nearby group of quiet picnickers to continue imbibing wine with their food).
It's also perfectly legal to consume alcohol on almost all National Rail trains (except where there's a banning order in place on specific services, such as those used by large groups of football fans). You can buy booze on many intercity services and M&S provide plastic cups in their 'Simply Food' food branches at railways stations specifically so that their customers can drink their wine (or whatever) onboard trains.
London Overground services are a little odd in that they're part of the National Rail network but run by Transport for London (who then outsource their operation to Arriva). So while passengers can normally drink alcohol on services operated by Arriva, and on almost all other National Rail services, the consumption of alcohol is prohibited within a small geographical area due to TfL's involvement. It's not hard to see how DA could have become confused!