spiritual relating to or affecting the human spirit or soul as opposed to material or physical things.
That is how many use the word, which is why it appears as such in a dictionary, but there are problems there. First, we don't know that anything exists which is not physical. As far as we know, there is only nature, and nature is physical, that is, matter, energy, and force dance through time in space. There may be nothing else, and we have insufficient evidence to conclude that there is anything else.
Let's not confuse material and physical. Physical refers to those five elements (matter, energy, force, space, and time), whereas material only refers to matter. The mind is not material. The brain is. But the mind is physical if it is a manifestation of these five and an epiphenomenon of the material brain. We don't know that that is the case, but we have no reason to think otherwise apart from a propensity of the mind to understand what I call the authentic spiritual experience - connection, warmth, belonging, transcendence, awe, mystery, and gratitude - in terms of spirits. There is a tendency for those undergoing such experiences to express that they have seen God, or that they have experienced some aspect of reality out there that is more than their own minds.
If that's the case - that the only reality is nature and that nature is physical - then "human spirit" and "soul," if they refer to anything, refer to aspects of physical reality, of brain and mind. The only things in my experience that those phrases might apply to is what I might call personality or human nature, which are consequences of brain activity.
It's been very helpful to me to understand reality and myself in these terms and to avoid the flights of fancy that come from attributing agency to these spiritual intuitions. I understand that experience as I do the sense of beauty or humor. When I find something funny or beautiful, I understand that that is a judgment of my mind in response to a certain type of stimulus or experience, and not a reflection of the world outside of my mind.
The Baha'i Faith recognizes that humans have two natures, a spiritual or higher nature and a material or lower nature, because that is Reality.
That's understood naturalistically in terms of the tripartite brain and mind, which results from the superposition of the mammalian brain onto the reptilian brain, and then the human brain onto that. The reptilian and mammalian impulses are survival instincts. We desire and fear. We desire food, water, air, mates, and to survive.
Absent the intellect, that means taking what we want without reflection or a bigger picture. But the human mind superimposes symbolic thought and higher purpose onto those and modifies them. The lion takes what it wants, including attacking zebra and mating lionesses their consent, and we don't judge it morally for that, but man has a higher nature that changes all that in those with a mature conscience.
We are cursed with multiple levels of mind generating conflicting impulses, which you've identified as base instinct and a higher nature. Freud called these the id and superego. Plato described a mounted rider, the horse being the unbridled urges and the rider being the bridle modifying and directing them.
nobody can prove that God exists except to themselves.
Which to me is a way of saying that one can convince oneself that gods exist in a way that they cannot show others, which is consistent with my description of interpreting spiritual experience as indicating a god.
Anyone who lived in my skin, would not need proof, but would know God exists, and angels, but not demons.
This tells me the cast of characters that you attribute to spiritual intuitions. Others such as the polytheists, interpret the same experience in terms of other agents. Some just say a higher force. I've eliminated all of these characters. There's just the experience and the good feeling it gives me. I'm not denying that there might be more. I'm just not willing to guess what that is, because I don't need to and find no value in guessing. Self-actualization is a type of intellectual, philosophical, and emotional evolution, and for me, that includes dismissing the cast of characters others impute to those intuitions.