(Answer was edited based on OP's edits and comments, see the bellow for the original answer)
You said that the orthodontist billed insurance $8k, so insurance paid based off of $8k (all is good).Then you got the bill from the orthodontist and they said the total amount is $10k (uh oh).
In other words the orthodontist charged you more than what they charged insurance. You should contact the orthodontist and have them explain why they have two amounts for the same procedure. You can even tell them "I called my insurance and they said you have a set fee of $8,000 for this procedure and that is what you billed them for. I will only pay my coinsurance based on that amount".
By the way,
The contract says that I am responsible for what the insurance does not pay
is 100% normal and does not mean the orthodontist is trying to screw you. This simply protects them from the people that don't understand coinsurance/copays and say "I have insurance so I shouldn't have to pay anything!". But, if the orthodontist is in-network and they charge you more than the contract amount, that's a problem!
In your case you can demand that you will only pay based on $8,000 since that's what they told your insurance the procedure cost.
The insurance company has a "book" of covered procedures and the acceptable fee for those services. If a medical provider is in the insurance company's network, then they agree to accept the amount given in the "book" as payment in full for the service. This is called the "negotiated", "contract", or "covered" amount.
If they are out of network
The insurance "book" says that your procedure should cost $8,000. But, your orthodontist is not in their network so he doesn't have to abide by that rate. So he bills insurance $10,000 and they say "we will only cover 90% of $8,000". This means you pay 10% copay on $8,000 and then the full amount above $8,000. Or
0.1*8000 + (10,000 - 8,000) = $2,800
If they are in network
The orthodontist can still bill as much as they want ($10,000) but then insurance will come back and say "you are in network and the contract amount for this procedure is $8,000". The orthodontist should then send you a bill like this:
Procedure cost: $10,000
Insurance negotiated amount: $8,000
Insurance payed: $7,200
YOU OWE: $800
The orthodontist can bill insurance whatever he wants, but if he is in-network then he must abide by their "negotiated" rate. Contact your insurance or login to your online account and figure out if the orthodontist is in-network.
If he is not, then everything is as it should be (assuming the math checks out).
If the orthodontist is in network, you should contact your insurance
and tell them an in-network provider is charging you more than the