We hear left-right, or horizontal. Height can to some degree be detected from changes in frequency response, but then you would need an object that moves in the height direction, because you need a reference.
Whether a speaker reveals itself as sound source is determined by properties of the speaker, and to some extent its acoustic surroundings, again depending on the speakers radiation pattern. Generally, a speaker that disappears well and image well for ordinary 2-ch stereo, will also disappear well used as a center.
It is true that a good 2-ch system can produce a soundstage where 3-d also extends into the height dimension, with instruments having different both location and apparent size in the height direction. But the speakers obviously have only one fixed height.
Like the horizontal plane, our ability to pinpoint a sound in 3D is aided not just our processing of the sound from a fixed head position, but processing of how what we hear varies
as our head moves. Once we add head rotation to the HRTF picture, perhaps you can see how we can locate a sound source vertically, even if it's frequency response is modified to 'emulate' a sound coming from lower/higher than it actually it: our ability to locate a sound by subconsciously correlating the HRTF's
relative changes in frequency response as our head's pitch angle changes is an algorithm that works for
any sound (of sufficient audible bandwidth), independent of that sound's frequency response. As to how obvious or subtle this is, I can't say (other than give personal anecdotes), but this clearly can't be ruled out entirely.
It's also possible that different people are attuned to horizontal and/or vertical imaging differently than others. For example, I've heard a good number of speakers over time, and
not once have I ever heard a convincing ability to separate sounds in space vertically -- only horizontally, and even then, barely more than one dimension. I'm not saying speakers can't project an overall soundstage that is wide and tall; rather, I am saying that this spaciousness is usually achieved uniformly, not in a way that allows one instrument to be placed really 'high altitude' while another is simultaneously really 'low altitude'.