They have always ignored its "official" name, choosing ones that meant more to them.
The second time, feelings were so strong, that the whole neighborhood ended having the same name.
St John's Gate and the nearby tower were built and named by the Knights, who also built here a church for the Saint. The locals however always called the gate "Koskinou Gate", because it was used by people from the homonymous nearby village to bring their products to town.
Things changed in 1522, when Rhodes was occupied by the Turks. It is said that so many people died fighting here, that the gate and everything around it were covered in blood. Ever since, the name "St John's Gate" only exists in books and tourist maps. 
People know the Gate and the neighborhood as "KÓKKINI PORTA": the RED DOOR.
They are honest, tough people, who can be loyal friends and fierce enemies. Poverty has been driving them away for centuries. Most men became sailors and quite a few of the families have emigrated.
However, the bond between people, the gate and the quarter is so strong, that wherever they may be, the only place they think of as "home" is KOKKINI PORTA.