Major surgery scheduled – need help to understand my insurance coverage

I have a major surgery scheduled for Jan 2023. I have health insurance through work, and I do not expect any changes to the plan for next year. The surgeon and hospital (1 overnight) are in-network.

In-network benefits –

Medical Deductible: $0

Medical Coinsurance: 0%

Out-of-Pocket Limit: $6,600 Individual/$13,200 Family

Hospital Admission: $1,500 Copayment

Inpatient Hospital Services: $1,500 Copayment

Inpatient Hospital Surgery: Covered in full

Does this mean, the max I would pay for the surgery would be $6,600 or $13,200? This is a family plan (I’m the policyholder). The message I got from the hospital says “Please be aware that there will be at least three components to your surgery charges. You and your insurance company will receive individual bills from your surgeon or surgeons, Hospital, and your anesthesiologist.”

It sounds like if they can’t get the amount they want from the insurance company, they will bill me separately. Am I correct? Is it possible?