Is this considered forgiveness, and do I lose my right to seek retribution if I forgive someone who wronged me to appease another party without feeling inner pardon for him?

If you voluntarily and willingly relinquish your right, then the pardon is valid, and the right is forfeited. You cannot retract it, for "what has been relinquished cannot be reclaimed." Pardon is commendable in Islamic law; it is a noble trait that Allah loves, a means to attain Allah's pardon and forgiveness, and a path to honor and dignity, in accordance with His, the Almighty's, saying: "And let them pardon and overlook. Would you not like that Allah should forgive you? And Allah is Forgiving and Merciful." And the Prophet, peace and blessings be upon him, said: "Allah increases not a servant in anything but honor by pardoning."