In Todd Akin's apology video, which his campaign dubbed "forgiveness," we find a handy example of the unprincipled variety I've been writing about.
He apologizes, but he declines to take pracitcal responsibility for the damage his remarks did to his cause, the Republican Party, and the voters he was nominated to represent.
Instead, he proceeds as if having said he's sorry, he's done all that can fairly be expected of a man. Hence, his "please forgive me," only adds to his original offense. It's as if he says to those he's just betrayed with his boneheadedness, "Now that I've apologized; it's your responsiblity to forgive me and move on."
To be genuinely sorry doesn't mean to feel really bad about what happened. It means to accept responsibility for what happened and to take the consequences for it (as much as possible) off the victim[s] and on oneself. "I did this; I own it. I'll pay for it."
Newman has a sermon called "Self-denial, the test of religious earnestness."
Likewise, actions are the test of contrition's earnestness. Leaving your victims to absorb the punishment for your wrong, while you blame them for "unforgiveness" is proof positive that your contrition is not sincere, but bogus and self-serving. It ought to be rejected as such.