Many procedures have bills from multiple sources, as you described. Basically do not worry about the source of the bill, just add up what you actually paid.  Whatever insurance did not cover you paid for "out of pocket". If you paid - it is an expense and you add it all up together. Premiums are what you pay to insurance companies, including for Medicare and "Medi-gap" premiums. Premiums are not medical expenses, however some premiums, such as if you are self-employed or pay for long term care insurance, are deductible.

The premium is the amount paid to the actual insurance company, Blue Cross Blue Shield for example, on a yearly basis (can be deducted from paycheck weekly or monthly) to provide you with health insurance. The premium has nothing to do with the doctors or hospitals.  The deductible is the amount that must be paid to medical professionals before the insurance payments "kick in".