Must a vow be fulfilled if its objective is achieved by making something different from what the one who made the vow intended?

If someone makes a vow contingent on a specific event, they must fulfill it if the contingency occurs, based on the Prophet's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." The intention of the vow-maker is sufficient to restrict the vow. Thus, if someone vows that if they are able to provide specific assistance to certain individuals, they will perform an act of piety, they are obligated to fulfill their vow upon gaining the ability to do what they specified. You should refer back to your intention and purpose. If you intended only the specific thing you defined, then you are not required to fulfill it unless that exact contingency occurs. However, if you only intended to bring benefit to those individuals, then you are obligated to fulfill your vow, for the rules of Sharia dictate that matters are judged by their intentions.