Words do matter. There is not a phrase actually called “accepted for value”. There is Accepted and there is Value. Because, I do understand the principle behind it though I can engage in the discussion.
The actual phrase that is better is this, “accord and satisfaction”. Why you may ask. Because, this phrase actually exists. It also is the correct term for the principle behind Accept for value.
accord and satisfaction (18c) An agreement to substitute for an existing debt some alternative form of discharging that debt, coupled with the actual discharge of the debt by the substituted performance. • The new agreement is called the accord, and the discharge is called the satisfaction. Cf. compromise; novation; settlement (2), (3).
“‘Accord and satisfaction’ means an agreement between the parties that something shall be given to, or done for, the person who has the right of action, in satisfaction of the cause of action. There must be not only agreement (‘accord’) but also consideration (‘satisfaction’). Such an arrangement is really one of substituted performance.” 1 E.W. Chance, Principles of Mercantile Law 101 (P.W. French ed., 13th ed. 1950).
ACCORD AND SATISFACTION, Black’s Law Dictionary (11th ed. 2019)