Sounds like you have all of the terms in your question. You want a full and complete release as against all parties. Get them to change the trade line, they will do it. Good Luck.
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Disclaimer: This answer does not constitute legal advice. I am admitted in the States of New York, New Jersey and Massachusetts only and make no attempt to opine on matters of law that are not relevant to those three States. This advice is based on general principles of law that may or may not relate to your specific situation. Facts and laws change and these possible changes will affect the advice provided here. Consult an attorney in your locale before you act on any of this advice. You should not rely on this advice alone and nothing in these communications creates an attorney client relationship. The opinions expressed herein are those of the author only and the fact that he has worked as an Assistant District Attorney; State Supreme Court Clerk; Special Assistant United States Attorney (Hawaii); Assistant Cornell University Counsel or Judge Advocate, United States Marine Corps should not be relied upon to assume that these statements reflect the policy of these organizations.
I think you've got the important terms covered. Yes, you should ask for favorable reporting on your credit. Even if they insist that they're required to report negative events and they're not allowed to delete true information, there's precedent, and legislative support, for lenders doing exactly this, e.g., when delinquent student loans are settled.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Debt collection settlements are best in writing, and the debt collection law firm or debt collector should be able to provide you with their standard form, so you can determine if the terms are acceptable and complete. It can be a challenge to get the debt collector to add or remove certain provisions. Some debt collectors refuse to put settlements in writing, but I cannot explain their rationale, as it makes no sense to me.
Many debt collectors will not delete a trade line, unless you have some evidence that it is inaccurate or incomplete, such as it shows an incorrect balance or is the result of identity theft or fraud. If you do settle the debt, they should then update the reporting to show a zero balance due and that you settled it. If you pay the full balance due, then it may report "paid in full." If not, then "settled." "Paid as agreed," should be used when the consumer has paid the terms as agreed with the original creditor. If you hold out for deletion, "paid as agreed," or "paid in full," I doubt you will be able to settle this debt with that precondition, unless you are willing to pay the full balance or it is a fraudulent account.
If you have any accounts or credit information that has been reported inaccurately, you can dispute this directly with the credit reporting agencies, regardless of whether you have this term in your settlement agreement. See link, below, to my web site with free sample letters that you can use to dispute inaccurate credit reporting. In particular, see letter 3.1. If you send it, be sure to provide your proof of payment and send it by certified mail, with return receipt.