Is the initial statement of donating to the mosque considered a vow that must be fulfilled, after having vowed and donated to a relative?

A vow (nadhr) is an undertaking of an act of devotion that has not been specified [as obligatory]. Its formulation requires wording that indicates commitment. The statement "I will give charity" is not considered a vow, but rather a mere promise that is not legally binding to fulfill, though fulfilling it is preferable. The obligation has been discharged by fulfilling what was vowed a second time.