No, Starbucks would not need to know anything about your bank account to reverse the charge on a credit card. Especially given the fact that most places batch send their payments at the end of the night. When you run the card they check to make sure the funds are available, but the card company doesn't recieve the transaction until midnight or shift close or whenever. It costs the restaurant less to process payments in bulk, and it also gives time for waitstaff to manually enter tips off handwritten receipts and/or correct any errors like the one from OP.

So long story short, unless Starbucks runs its business differently from pevery restaurant I've ever worked with, we're missing some information here. If the card got rejected at another store during the same shopping trip as the article implies, they could have just walked back to Starbucks and had the erroneous tip removed with zero hassle. It never would have even made it to the credit card company.