Nothing is'supernatural', but there are things which, while perfectly natural, are deemed such because we do not or cannot understand their nature. Why and how do certain experiences or events resonate enough to produce echoes that can be seen/heard/felt years or centuries later? We don't know, it goes against the 'laws' we have constructed for what we call 'nature', but we don't know all the laws.

Are the 'presences', like the one I felt so strongly in the woods near my University, supernatural, or merely products of another level of nature we are usually unaware of?

Paranormal? Some people seem to have an innate ability to perceive the quantum flux and foresee the probability of something happening. Others have the ability to achieve a kind of quantum entanglement with others that allows them to know what they might be thinking. Still others are more sensitive to the echoes people and events lieave behind them. We call these people 'fey' or 'psychic', I borrow a term from WH40K and call them Psykers. Setting aside the obvious charlatans, there are still people out there who have these abilities. They are no more 'paranaromal' than green eyes, extra fingers and toes, or third nipples. They are just rare, and people who have them don't always choose to shout about it.

'Natural' and 'normal' are the words to be worried about, especially the latter. Beware the person who tells you this or that is 'natural'. They're probably trying to sell you something. As to a person who claims to be 'normal' - they aren't, and neither is anyone else! 'Normality' is a social fiction designed to ensure we don't show off too many of our individual quirks in public and scare people who have different quirks!