Ok, let's throw away the term "I Am" then, I've never liked that one anyway.
There is a tone that is "heard" at all times but not with the ears, I don't always notice it when I'm busy but it is always easily accessible and very noticeable when I'm quiet. There is a static-y, sort of sparkly thing that happens in the visual field that can also be found at any time but is most obvious when the eyes are at rest.
I experience both of these, and they are more apparent the more quiet the mind is. I wouldn't attribute them to any kind of presence though.
Sometimes the presence is felt like strong waves of energy moving through the body, the head drops, the eye close, taking in the feeling of presence.
I feel these strong, goosebump-like, waves often. Again, it happens more when the mind is quiet. Why would you describe it as presence or the feeling of presence, rather than the sensation that it is?
So, it seems like you experience what you call presence at times when things are quiet, but could you be reifying and defining presence beyond what you experience it as?
It is certainly not separate from sound or other senses, it feels like the medium of sound and senses.
How can it feel like the medium of the senses? What does that feel like? Does it really FEEL that way or is that a guess, a thought? The senses are dull when one is lost in thought, but everything is much more vivid, luminous, alive, and "present" when the mind is quiet. But is vividness, aliveness, luminosity, or presence palpable? Are they any more than labels used to describe experience like "loud" or "blue" or "rough"?
Taking a guess here, but I'm sure you've heard and read a lot of spiritual content about some kind of essence or presence or awareness being the fabric/medium/container of everything you experience. Investigate your experience and see if this is experientially the case. Would this concept occur to you to describe presence if you had never heard of the concept before?