Okay, bear with me.
Medicare part A is basically hospital care. Most people qualify for it at age 65, and usually it's free if you've paid into the SS system long enough, or your spouse has.
Medicare part B
is a bare bones medical insurance policy that you
must sign up for if you want it, and requires a monthly premium. Part B "can include outpatient care, preventive services, ambulance services, and durable medical equipment. It also covers part-time or intermittent home health and rehabilitative services, such as physical therapy, if they are ordered by a doctor to treat your condition."
https://www.ehealthmedicare.com/about-medicare/medicare-part-b/
Part B does not include a drug plan but you can either buy drug coverage through a private company, a Medicare Advantage Plan, or forego it altogether, but if you don't get one as soon as you first sign up, you pay a penalty thereafter. You can buy
supplemental part B plans, called Medigap through private companies, to improve the care and coverage of your Medicare part B insurance. These plans use your part B payments as
part of their cost to cover you, and may charge a monthly premium in addition. That's why you continue to pay your part B payments--
your private company won't cover you unless you are signed up for Medicare part B, and that's why they are referred to as Medicare supplemental plans. These plans are regulated by the federal and state governments and there are currently 10 different options for them, "the benefits for all the basic benefits options are generally the same regardless of insurer. The differences will be in the price, who administers the plan, and which of the 10 options the insurer chooses to offer."
A Medicare Advantage Plan (part C) is a medical insurance plan, offered by a private company, that must offer at least what original
Medicare parts A&B offer, plus a drug plan. These advantage plans take your Medicare part B payments as part of the cost of your coverage, like the other plans, but include the drug plan as well. These plans coordinate the coverage for parts A and B, as well as providing other benefits like dental and vision. They can be in the form of HMOs, PPOs or other organized medical provisions offered by private companies.
The OP sounds like he purchased an
Advantage plan. A Medigap plan covers the "gaps" in costs for services that Medicare doesn't cover, so the doctor shouldn't reject that card or ask which to use. If the OP had an Advantage plan, however, that is the plan that covers all benefits and is the one that would be accepted over the Medicare card.
Use the link in my post to clarify anything that still seems confusing.