I think there is a simple answer to the meaning of natural/unnatural. It refers to the criterion provided by nature: the feelings.
A natural desire is one that is likely to result in net pleasure if fulfilled.
An unnatural desire is one that we only imagine as likely to produce pleasure, but in fact is likely result in net pain. Also referred to as "vain and empty". The first definition that comes up when I search "vain" is "not yielding the desired outcome; fruitless" - the desired outcome being pleasure. Empty means empty of pleasure.