I think this would be a difficult to one to nail down, as most people essentially experience the world completely differently, whether within music or not. Even if the response of the brain is the same for the different people who use the same types of memorization techniques or report having the same visual experience, it's hard to say that they're all having the same experience. How do I know that you and I experience the same thing when we look at the color blue? We may both see the same object and respond to it saying it is blue but for all we know, my brain is perceiving the colors differently. So I think this answer would be different for everyone but I imagine there are some common experiences.
What I can speak to is what a friend of mine who is completing his doctorate in piano performance has told me. One of his teachers had taught him good practices for memorization that involved a few steps. One was to memorize what the sheet music looks like and be able to recall it well enough to write out the notes again, so he could see the score in his head while he played. Another way was to just memorize the music itself, being able to sing the melody in your head and audiate/conceptualize the rest of the music (since you can't really sing chords). Another is to memorize the physical movements, ie, which keys you press when and with which fingers, when to shift positions, etc. All of this comes together to form the overall memory but can also be used as a crutch if the player becomes unsure of what to do next, so if they can't remember, they can visualize or audiate any aspect of the music or performance to get back to confidence. So they might visualize the score or physical motions, or they may just audiate it with no visualization.
The thing that really comes to mind for me though is that I've heard many people, as well as having experienced it myself, report that they are performing but actually thinking about something completely non-musical, such as what they have to do the next day or how beautiful the girl in the front row is. This is particularly interesting for people who are singing, especially if they play an instrument while they do so. The singer in my band has related to me how she has been playing guitar and singing through the verse or chorus and had the realization that she has no idea what is happening next in the music, then racking her brain trying to figure it out. Next thing she knows, she's playing and singing the next part without having even realized it. All of this speaks to what most refer to as muscle memory but is really called procedural memory. So with enough practice, no thought is even required, let alone visualization. This is also exemplified by someone who is blackout drunk pretty well (which I've done a couple times when I was younger and strongly discourage), though a little different.
I would also think that this varies greatly depending on the experience of the player, the difficulty of the music and the style of music. A beginner is almost certainly thinking about how to execute each note or chord, with the only visual stimuli being them staring at the instrument. Someone playing Jazz may be visualizing the chord changes as they improvise. Someone playing the most difficult music you can imagine might be visualizing the score. Some people just don't "see" anything.
Beyond all this, there is also synesthesia, which is basically when a person's senses cross, the most commonly discussed of which seems to be seeing colors when hearing notes or music. These people are clearly seeing something entirely different than those that are not synesthetes. Research shows that the visual stimulation that is experienced is entirely subjective, ie, there is no reliably common visual representation of the music from different synesthetes.
So I think it's ultimately different for everyone, even when it's reportedly the same. With that in mind, unless someone has an answer based on research with references to experiments or surveys, this question is a bit outside the domain of this forum, which is kind of a shame, as it is an interesting topic, but I understand and agree with the reasoning.