It is my understanding, that some plans don’t have a format “redeposit” process - some do (e.g., Fidelity).
The IRA doesn’t mandate a special process - so using a standard contribution would fulfill the rules.
However… the IRS only allows “redeposit” for refunds (e.g., when a class was dropped, etc.). In fact, any redeposit form might actually make you declare and sign that this IS indeed due to a REFUND - to comply with IRS rules.
If you withdrew more than the actual expenses you had incurred (e.g., for some reason you withdrew without having incurred expenses) - then the penalty applies. Making an equal-amount contribution won’t undo that.
PS - do you have the option to already pay the Spring Semester? Colleges can bill you in December for the next calendar year, and these amounts will be appropriately declared.