The churches in our denomination never agreed on whether to say "debts" or "trespasses," when praying the Lord's Prayer in unison. When we are worshipping with sister churches and the leader doesn't tell us which to use this time, God no doubt hears our meaning, but the rest of us hear "forgive us our mumble." I served five of our congregations as the duly installed pastor; they were split three to two between debts and trespasses.
When he gave us this prayer, Jesus said "debts" and "debtors." In this dispute however, tradition, or local habit, or whatever, trumps scripture: "Jesus may have said debts but around here we don't!" (Substituting "sins" for debts is well-intended, but makes visitors to your church feel like they don't know what is going on. It's a less precise translation anyway.)
I'll use either word, and try to go with the flow wherever I am. But whichever word I use, I want to mean what Jesus meant.
Debts and trespasses are almost interchangeable, but not quite. All debts are trespasses, but not all trespasses are debts. Debt applies to more than money, and trespass applies to more than real estate. My working definition is that debts are trespasses which incur actual harm by impeding or distorting what God intends for ourselves and our world. They cause real damage that alters the state of things.
Saying "That's okay" is not a fix. Lesser trespasses may be occasions for discipline, excuse, or pardon; but these require more: that the debt be paid, the harm repaired, the loss restored. Forgiveness is neither excuse nor pardon, but what happens when the victim fully and willingly bears the pain, loss, and cost without holding the trespasser accountable. Jesus called it forgiveness from the heart.
So when we say, "forgive us our trespasses," we are not saying, "please excuse us, because nobody got hurt"; rather, we are begging God to take the hit without requiring payback. When we dare add, as Jesus commands us to, "as we forgive those who trespass against us," we are claiming not only that we are able to do that, but that we actually do it.
We are always able to take the hit. I heard Canadian pastor Walter Boldt put it this way twenty-five years ago, and have never heard it better said: to forgive you must give up your rights. You are owed, but you don't collect! You take the hit. It is not hard to do, because you don't have to do anything. You only have to suffer it—just like you're asking God to do for you, when you ask him to forgive your trespasses.