This doesn’t mean that they are unsociable or always keep everything secret. They just love to do things by themselves to feel as if they don’t need anyone else doing this for them. Is there any word or phrase that can interpret this full meaning?
Self-sufficient has the connotations of not needing others.
Especially in the second meaning from Dictionary:
having extreme confidence in one’s own resources, powers, etc.
self-sufficient, and always reminded you of it.