US bank deposits over $10K only need to be reported to FinCEN (Financial Crimes Enforcement Network- a bureau of the US Department of Treasury) if the deposits are made in cash or other money instruments where the source cannot be traced (money orders, traveler checks, etc). Regular checks and wires don't need to be reported because there is a clear bank trail of where the money came from.
If your family member is giving you money personally (not from a business) from a bank account which is outside of the US, then you only need to report it if the amount is over $100K. Note, you would need to report that regardless of whether the money was deposited into your US bank account, or paid directly to your credit cards on your behalf, and there are stiff penalties if you play games to try to avoid reporting requirements. Neither deposit method would trigger any taxable income for the scenario you described.