Who actually killed John Howlett in the Wolverine Origins Graphic Novels

After the cage fight with Smitty Logan walks off and then Dog finds him and they fight. Dog says "First I gotta see if little James remembers how i was the one who killed his Poppa" Then Logan says "you killed my father you monster.....all these years I thought it was me"

I don't understand as it was Thomas Logan who shot John Howlett Wolverines "dad" and it WAS Wolverine who killed Thomas Logan his biographical dad. It couldn't have been Dog.....Can someone help me understand this