How can the Hadith "Free her, for she is a believer" be reconciled with Sa'd's Hadith, which indicates that a person is described as a Muslim based on outward actions, knowing that the slave girl's speech is considered an outward action?

It is permissible to describe a specific person as a believer (mu'min) if it is to indicate their inclusion in the apparent rulings pertaining to believers, such as freeing a believing slave or the prohibition of harming a believer, but not for the purpose of commendation (tazkiyah).

Shaykh al-Islam Ibn Taymiyyah differentiated between apparent faith (iman zahir), upon which worldly rulings are based, and inner faith (iman batin), for which a servant earns reward in the Hereafter. The Hadith of Sa'd with the Prophet, peace and blessings be upon him, when he asked about a man, "Or a Muslim?" illustrates that testifying to apparent faith is different from testifying for commendation.

Scholars have agreed that applying the term "believer" to oneself or others for the purpose of commendation is impermissible, except for those whom the texts have explicitly attested to.