I've never heard the term "linear loading", and searching for it I can only find amateur definitions, many of them contradictory.
Many reference J-poles and some reference (although they don't use the words) folded monopoles. What these have in common are transmission line stubs with the objective of modifying the feedpoint impedance.
Calling parallel wires in proximity to other wires "loading" is something to be avoided, in my opinion.
Loading, as with a loading coil, or a capacitive hat, are techniques for changing the electrical length of antenna, usually to make it electrically longer than it is physically. A loading coil at the base, or a capacitive hat at the top of an antenna has a predictable effect: it makes the antenna electrically longer. Since the effect is predictable and the technique commonly used, generalizing these techniques as "loading" is a useful terminology.
However, adding some parallel wire can have myriad effects. It might be mostly like a transmission line (like in a folded dipole) or it might be a parasitic element (like in a Yagi). It might change the impedance seen at the feedpoint (in any direction), or it might make the antenna more or less directional, or it might just be a balun which might be designed to avoid altering the impedance and radiation pattern from the idealized free-space case. The function of some parallel wire will depend on the spacing, the relative diameter, whether it's short, open, or something else at each end, where it's connected, and many other factors.
Since the effects of some parallel wire are so diverse, and there are so many possible variables, and there appears to be no solid consensus or professional literature to establish a definition of "linear loading", I would not any kind of parallel wire "loading" if my objective was to communicate clearly. Moreover, in some cases such as the J-pole the function of the parallel element is not "loading" (changing the electrical length of the antenna) but rather matching: the radiating element of the J-pole is already the desired electrical length, but the impedance is too high. "Folding" as in folded dipoles and monopoles is pretty well defined, but these designs rely on the parallel elements being shorted at the end farthest from the feedpoint, which seems to be a different thing than you are asking.
As for your question about an antenna with another wire nearby and what it might be called, I'm not aware of a name for any such thing. As explained, the behavior of such an arrangement would depend quite a bit on just how far apart the wires are and their lengths. I think a professional would call this "a low dipole with a wire buried under it", and the properties of such an antenna would be determined by modeling or empirically measuring such an arrangement of the particular geometry under consideration.