Is it considered haram to give money in exchange for someone buying something for me, because it falls under the ruling of a gift?

Your action in this regard could be considered a reward or a reciprocation for their gift. Accepting a gift and reciprocating it is better, as the Prophet, peace and blessings be upon him, said: "Exchange gifts, and you will love one another," and "Whoever gives you a gift, reciprocate it." Rejecting it is permissible, because accepting it is a sunnah (recommended) and not obligatory.