If you still owe money on the car, the lender is not a bystander. The lender holds a lien on the vehicle, and it may have its own rules about protection products that touch the car. A service contract is not the same thing as an insurance policy, and the rules vary by lender and by contract, so do not assume the lender's terms apply or do not apply. Ask the seller and the lender directly, and get each answer in writing.
Who gets the refund if the loan is paid off
Some contracts can be cancelled before the end of the term, with any unused portion of the price returned under a method the contract spells out. When a loan is paid off or refinanced, the refund may be sent to the lender, to you, or to a third party, depending on how the contract was financed and what the lender requires. Ask who the refund is payable to, which section says so, and whether a payoff letter changes that answer.
Does the lender need to be named
Ask whether the lender is listed on the contract, and whether that listing gives the lender any right to file a claim or receive notice. A lender named on the paperwork may get copies of notices you never see. Ask who receives copies of cancellation notices and claim decisions, and ask whether you can change that list later.
Financing the contract into the loan
If the contract price is added to the car loan, the cost sits inside your loan payment. Ask what happens to the loan payment and the loan balance if the contract is cancelled early, and get that answer in writing. Ask also whether the contract can be paid separately instead. Keep both sets of documents, because the lender and the provider may each hold a different version.
Use the table to compare the paperwork
When you compare providers, look at how each one treats cancellation, transfer, and lender notices in the terms, not in the sales pitch. This site's vehicle service contract companies compared, term by term page lays the published terms side by side. Use it to find the sections worth reading, then confirm each point in the contract you would actually sign.