Names are always contextual. In the context of a live view module the %Phoenix.Socket{} is never accessible, so you don’t need to distinguish the live socket you have from an ordinary phoenix socket. In a similar way, we always refer to a %Plug.Conn{} as conn even though there is also a cowboy connection out there making the request work. The cowboy connection isn’t really accessible in the context of a plug, so conn has plenty of context and there’s no ambiguity.

Right, if I have a module that has helpers and some are for live view and some are not, then in THOSE functions I would have the variable be named live_socket. The context of the helpers is different from the context of a live view module. The caller of a function and the function argument don’t need to be exactly the same name. Type specs can be very helpful there too.