The bank is free to lend out $9,000 to another person, and once that’s deposited, the bank can lend out another $8,100 to someone else, and so on and so forth.

The bank was free to make that loan before you came along. Your deposit has no relevance whatsoever. If the bank is obliged to have $10k in reserves for every $100k it lends, it can find that $10k anywhere, including borrowing it from another bank The cheapest way for the bank to find $10k is to offer you checking services on that amount at a very low price. But no one makes you lend the bank money in exchange for checking services, and the bank has not "lent out" your money. The bank has extended its own credit, and it has met its reserve requirement by bartering services with you instead of selling stock or debentures or commercial paper.