Does the Sharia consider a man to be a husband to a woman or merely engaged to her if an agreement on the dowry has been reached in the presence of witnesses, even though this agreement is usually considered a custom and not a legal contract?

Marriage has pillars: that the woman's guardian or his representative contracts it, with two just witnesses, and a formula indicating the contract. What happened was merely an agreement on the dowry in the presence of witnesses, and this is not sufficient for the conclusion of a legally valid marriage, which requires a formula of offer and acceptance. Therefore, the questioner is not considered the woman's husband merely by virtue of this agreement.