What is "being out of job for a personal reason" called?

Solution 1:

As in this article:

1999 Washington Post (Electronic ed.) 18 June In a world where everyone has sick days, personal leave and mental health breaks, 12 years of perfect attendance is a rare feat.

Personal leave if frequently referenced in AmE to absences from work due to 'personal' reasons, which you may or may choose not to elaborate on.

Solution 2:

Presuming you want to put the best face on things, compassionate leave and family leave seem best of the alternatives you listed, even though as noted in a comment you were not actually on a "leave".

Career break as described in Wikipedia seems neutral and applicable, although three years is a longer term than that entry mentions.

If you were an actor, the euphemism resting would apply. If you were previously employed and returned to the same employer, sabbatical might be appropriate. Other terms for gaps include hiatus and interregnum, the latter more applicable if you are a government or royalty. Or you could say you were slacking, but some regard that as not good on a resume.