Is what happened between the two parties considered a legitimate marriage contract, bearing in mind that the questioner's intention was only engagement?

If the phrase "give me" is used in the context of marriage and only betrothal is intended, then no marriage is contracted thereby, because it is not among the explicit wordings of marriage. The jurists have adduced evidence that marriage is not contracted except with the two words "al-inkah" (to give in marriage) or "at-tazweej" (to marry off), and anything other than these does not validate the contract, even if the intended meaning is marriage. They supported this with what Al-Shafi'i said: that Allah named marriage in the Qur'an as nikah (marriage) and tazweej (marrying), and that the private parts are forbidden before the contract and are not made lawful except with the two words at-tazweej or al-inkah. Marriage is not contracted by the word "gift" except for the Messenger of Allah, peace and blessings be upon him. Since this wording is not explicit, the woman remains a stranger, and it is not permissible to seclude oneself with her or to look at her. It should be noted that the guardianship of the brother takes precedence over that of the paternal uncle.