Is an intimate relationship with a Christian girlfriend, with the approval of family and friends, considered adultery despite the absence of an official marriage?

If you have entered into a valid Islamic marriage contract with the woman, then she is your wife. Otherwise, you both are engaging in zina (unlawful sexual intercourse). A valid Islamic marriage contract requires an offer and acceptance, such as the woman's guardian saying, "I give her to you in marriage," and your reply, "I accept." The mere knowledge of her guardian about your relationship and its widespread awareness does not constitute a valid Islamic marriage contract; rather, it is zina, and you must repent from it.