Is it a condition to inform the litigant of one's heartfelt forgiveness towards them to lift the sin of contention mentioned in the Hadith of the middle of Sha‘ban night, or is heartfelt forgiveness without informing them sufficient?

Since you greeted her and forgave her in your heart, nothing more is required of you, and the least by which abandonment ends is the greeting (salam). As for someone whose association harms a person's religion or worldly life, their connection or communication is not obligatory. You are not among those included in the lack of forgiveness mentioned in the Hadith.