I believe that the position that I want to become is called a gameplay programmer, but I don't know if this is a specialized role, or if this is just a small part of the responsibilities that are split between programmers. It probably depends on the studio size and if it is a AAA developer, but can this be something that I can specialize in and only/mostly deal with? Thank you.
network, render, engine, UI and gameplay seem to make up the main discipline split (not counting support stuff like build engineers and online service types)
Realistically in most small or indie studios you're not going to see significant engine, network or render work as you'll likely be on a third party engine and most of the coders will be dealing with Gameplay or UI.
Gameplay itself is a wide field. There's a whole load of specialisations within it - exactly what they are will depend on the project and most people seem to deal with an vague area of specialisation (eg, player movement, combat systems, AI etc) while still being fairly flexible