Here is a thought -- maybe the Hurt Doctor's crime was resurrecting Rassilon? This would fit -- All the NuDoctors have owned up to ending the time war. However in EOT, Tennant's Doctor said that the Time Lords have become ruthless and power hungry (or something to the effect). Maybe the trigger for their fall was bringing back Rassilon. Hurt could have brought him back, sacrificing a regeneration in the process. This would allow Eccleston's doctor to end the time war. The "shame" comes not from the act of genocide of the daleks and already fallen timelords -- an act which likely had no possible recourse. Rather shame is from the knowingly and actively introducing a corrupting influence into Time Lord society, causing their fall.
In the course of the program, the Doctor has reluctantly killed and allowed others to die. He has even caused many others to die through his actions, inactions, errors. The doctor fights EVIL not DEATH. Everywhere he goes, he opposes tyranny, oppression and corruption.
How often do we see the Doctor take a deliberate stand FOR the establishment. FOR the oppressors? In Logopolis, we saw a good chunk of the universe be destroyed because he made a mistake and led the Master to Logopolis. By now, collateral damage is old hat to the Doctor.
Of course, this is only speculation, and we simply do not know. They may avoid the Time War altogether.