The legal term that describes what you are referring to is called an "accord and satisfaction" and is covered under Mass Uniform Commercial Code which states:
Section 3-311. (a) If a person against whom a claim is asserted proves that (i) that person in good faith tendered an instrument to the claimant as full satisfaction of the claim, (ii) the amount of the claim was unliquidated or subject to a bona fide dispute, and (iii) the claimant obtained payment of the instrument, the following subsections apply.
(b) Unless subsection (c) applies, the claim is discharged if the person against whom the claim is asserted proves that the instrument or an accompanying written communication contained a conspicuous statement to the effect that the instrument was tendered as full satisfaction of the claim.
(c) Subject to subsection (d), a claim is not discharged under subsection (b) if either of the following applies:
(1) The claimant, if an organization, proves that (i) within a reasonable time before the tender, the claimant sent a conspicuous statement to the person against whom the claim is asserted that communications concerning disputed debts, including an instrument tendered as full satisfaction of a debt, are to be sent to a designated person, office, or place, and (ii) the instrument or accompanying communication was not received by that designated person, office, or place.
(2) The claimant, whether or not an organization, proves that within 90 days after payment of the instrument, the claimant tendered repayment of the amount of the instrument to the person against whom the claim is asserted. This paragraph does not apply if the claimant is an organization that sent a statement complying with clause (i) of paragraph (1).
(d) A claim is discharged if the person against whom the claim is asserted proves that within a reasonable time before collection of the instrument was initiated, the claimant, or an agent of the claimant having direct responsibility with respect to the disputed obligation, knew that the instrument was tendered in full satisfaction of the claim.
So if you were acting in good faith in tendering the check, and it stated "full payment" and they didn't return your payment within 90 days, then the debt is considered settled.
But if you just said "payment for past and present dues", that doesn't say you are making full payment so the creditor could claim that since it doesn't say "full payment" then it is just a partial payment and there is no accord and satisfaction.