Either forgiveness is subjective, meaning something different to each person, or it is objective, with a coherent, non-contradictory definition of what it is in its essence. If forgiveness is subjective, then there is no need for me to answer this question because it is “different strokes for different folks.” Yet, your question suggests that you see that an objective answer exists. There are over 800 books now in print about this topic and I have to presume that each author struggles to bring forth a true definition of forgiveness, otherwise why write the book?
So, it seems intuitively obvious that forgiveness is objective with an essence to it, which means that it has a meaning apart from other similar constructs such as tolerance, legal pardon, neutrality, indifference, mild annoyance, and “moving on from an offense.”
Given the objectivity of forgiveness, what does the term “to forgive” mean? It cannot be both a moral virtue and only thought control to aid oneself. Why? Because no other moral virtue is exclusively about oneself. Virtues flow out of one person to others for their good. If we insist that forgiveness is not a moral virtue, then it is imperative that those so insisting tell us what it is (and break with about 3,500 years of thinking on this matter).
For now, we are safe in assuming that forgiveness is a moral virtue. Thus, if it is, then it cannot—absolutely cannot—be defined as the cessation of resentment for an offense. Why? Because I can demonstrate tolerance and cease to resent. I can demonstrate indifference, and mild annoyance (without the emotional depth of resentment), and even “moving on” from an offense and cease to resent.
So, how can we distinguish forgiveness from all of these other ideas? We do so by defining it in such a way as to honor the “moral virtue” aspect of forgiveness. All moral virtues involve goodness toward others. What is the goodness that forgiveness offers? When a person forgives, he or she deliberately offers the goodness of understanding, kindness, respect, generosity, and even love toward the offender.
Of course, people need not completely fulfill this definition to be forgiving. We all fall short of perfection in expressing any virtue. Our human imperfections do not invalidate what forgiveness is.