Is my oath considered valid or void regarding the permissibility of residing with my second paternal aunt, who did not explicitly state her permission, but whose actions indicated her consent, keeping in mind that I swore a legitimate oath that they would permit me to reside there?

The most absolving for the questioner's conscience would have been to swear before the judge regarding the situation she mentioned, namely, that her paternal aunt did not object to her living in the house and did not demand rent. This would suffice to refute the aunt's claim for rent for the previous period, and the questioner's residence in her aunt's share would be considered a gratuitous loan (عاريّة).

Regarding the ruling on the oath: If the questioner swore this oath believing herself to be truthful based on what she understood from her aunt's permission for her to reside, then this is not a false oath (yamin ghamus), nor is she an oath-breaker, and there is no expiation due from her.