Some people (estate agents come to mind) referring to London Overground as 'Ginger Line'. This one annoys me the most by far.
Locals around Colchester - I've heard 'North Station' too, and although it's not the official name now, it is useful in conveying that the station isn't very near the town centre. The other one I don't think I've seen mentioned here is Colchester Town is often referred to as 'St Botolph's'.
During my time working at stations I heard the St Mary's Cray... I get that, like someone else said on here, St Paul's Cray is close to that station.
Shortening the name can be confusing: "I want to get to Wellin" = I thought it was Welling (nearby) but to the customer's dismay, they meant Welwyn Garden City! I also had a few tourists in my time who would ask for a ticket to e.g. Leicester or Worcester, for them to be outraged that a ticket to a tube station costs £90.
Holborn Viaduct - I understood what they meant fortunately, and the person told me it was their first train journey in 30 years. To be fair I only knew this as an enthusiast, and this sort of information should be provided in training.
I understand the use of British Rail/Railways to an extent, as well as "the Overground", "Tfl", "Network Rail", however in a customer service setting it definitely makes communication difficult to ensure everyone in the conversation understands what is being referred to. This is where I think it's best to stick with brand names. Ticket validity and stopping patterns, journey times can vary according to the TOC. It also creates confusion around complaints - I received a lot of complaints about other train companies' services or TfL issues that I literally physically couldn't resolve because "well, aren't you working for British Rail/Network Rail/TfL? Well what do you even do then?"