Hello,

As per your requirement, i would say that tge criteria to be called as private student is, 

Private Candidates:

Means the person is not actually a regular candidate but under the provison of laws and is eligible according to the rules is allowed to give the exams.

Example-

Lets suppose you don't go to school or are officially entrolled in the classes,but are still allowed to give the exams,also you have to study on your own,Then,you will be called a private candidate.

According to CBSE,Private candidates are not allowed to give compartment or improvement exams unless and until they fulfill the eligibility conditions which are mentioned on the official site of CBSE.

Yes,According to the latest changes made Private Candidates are eligible for NEET.

For more details,you may go to the link below:

Https://medicine.careers360.com/articles/neet-2018-what-nios-students-and-private-candidates-should-know

Hope this was helpful!

All the best!

Good luck