The key thing about the suffix "네" is that it indicates the family or some other group the person belongs to (a plural). 지네 and 자기네 are about the same, and both mean some group of the person referred to as 자기. As you mentioned, -네 form is also used as a possessive noun/pronoun. For example, 지네 집, 자기네 학교, 걔네 친구들, 얘네 서클, 쟤네 형제, 철수네 가족. But it is also commonly used as a subject, and sometime even as an object (쟤네 참 착해; 나 걔네 잘 알아). So 지네/자기네 is not the same as 자기의. They're different in meaning and have the plural vs singular distinction. Also note that 자기 by itself very often acts as a possessive too, especially colloquially. So "자기 돈" and "자기 목표" is much more common than "자기의 돈" or "자기의 목표", for example. Korean pronouns, especially the colloquial ones, are really complex because there are so many different variations. And the text book favorites 그, 그녀, 그들 are practically never used in real conversation, making things even more confusing :-)