I understand that 世帯主 means head of household but what specifically defines 世帯主 listed in 住民票? Is it a special legal term that includes certain rights and obligations?
住民基本台帳法 has a line requesting to nominate a 世帯主, but no definition about who should be and what it is. Only an administrative manual (住民基本台帳事務処理要領) without legal status issued by MIC has a brief definition.
A household (世帯) is a unit in social life that shares a housing and livelihood. Among the member constituting the household, who preside it is the head of household (世帯主).
In other words, there's practically no hard criteria to be 世帯主. Speaking in extremes, if all the member agrees that you are 世帯主, you will be.
That said, the status of 世帯主 gives you a little privilege to perform some formalities on the behalf of other members. It also affects tax calculations, public insurance, or other public services to varying degree. The system is generally designed on assumption that 世帯主 is who earns the main income of the household.
For details, it's better to ask your local government or legal experts.