There are at least two common meanings for student
- a person following a course of study, as in a school, college, university, etc
- a person who makes a thorough study of a subject
The former is the more common usage in the US, and I believe elsewhere. The latter, which can refer to informal, self-guided study, is almost always followed by the subject.
The President is a serious student of economics.
If the term is used without a subject, most people would assume that the person being described was in a formal education program.
Also if there is no specific information about school attendance (or lack of attendance), people would generally assume that the described individual was enrolled somewhere.
AS to what to call yourself, I would suggest self-guided student.