This sounds like a programming question, so I'm answering as a programer with more technical information, hopefully I'm right about your intent.
Public vs Private: the most common terminology to distinguish between what you use personally, and what you are making available to others. Though generic enough that it's precise meaning varies by context and even language used.
Internal vs external: also commonly used phrases. external would be your users, in a game case the gamers. Thus internal would be all the things the developers know about, but don't intend to share to the rest of the world.
API: definition of commands that can be used by another. In essence your API will effectively be a list of all your public facing objects and methods; everything you want someone else to know about and use, while anything not in your API is implicitly assumed to be private.
calling something 'base' also implies the second half. A base object implies an object that is intended to be expanded on by other objects later. Technically a base object could exist in the game itself, but usually the presumption would be that the base object exists only to be extended; and thus it most likely will not be exposed to the regular game or players directly.
You can of course combine these terms. Saying a private base object would fully define what your saying. Languages may have specific terminology to them you can use as well. In java terms you would say "private abstract class" for instance.
You could also use abstract or interface to have similar meaning as 'base'. but those two terms are a little more technical and have slightly more explicit meanings.
on a more general term 'mechanics' is usually used by gamers to reference the nitty gritty of how a specific object, or more commonly class, ability, or item, functions. actually when someone references the 'mechanics' of an object they are usually referring to the exact math or rules behind an object. In essence their saying 'the stuff that is knowable or calculatable by avid players, but may not be obvious or made readily available to every day gamers'. So every gamer may be told this spell does damage, but when someone discusses the mechanics of a spell they may be discussing specifics of how the spells damage is calculated. Thus a discussion of mechanics can be somewhere between the two points of view you mentioned, not always known to all gamers, but not impossible to access by those who wish to know.