What is the definition of a betrothal (khitbah) in Islam, and is it permissible for a betrothed woman to go out with her fiancé to the market or travel with him for a short distance (a few kilometers) using public transportation?

According to Sharia (definition), engagement is the suitor's request for marriage from the prospective bride's family. If accepted, it becomes impermissible for another to propose to her. As for your brother going out with the woman, if a valid marriage contract has been concluded (with offer, acceptance, and two witnesses), then she is his wife, and it is permissible for him to go out, travel, and be in seclusion with her, while observing customs. However, if it is merely an engagement, she remains a stranger to him, and it is not permissible for him to go out or be in seclusion with her, nor to enable him to look at her beyond the permissible gaze for engagement.