Is a ball intercepted at one's own one yard line then fumbled out of the back of the end zone a safety or a touchback?

Neither. Consider this situation: Team A is backed up to its own end zone. Team A's quarterback throws a pass that is intercepted by Team B just in front of the goal line. The Team B player tries to run around a couple Team A guys to get into the end zone. In the process he gets hit and fumbles the ball. The ball sails out of the end zone and out of play. Because the ball was fumbled and then went out of bounds, Team B retains possession and has a first down at the spot of the fumble. There was a game in 2001 when Patriots receiver David Patten was hit by a defender and knocked out cold, fumbling the ball in the process. The ball shot straight up in the air and came back down, bouncing off Patten's leg, and was recovered by the other team. but because part of Patten's unconscious body was lying out of bounds, by rule that meant his whole body was out of bounds, so when the ball hit his leg it was technically landing out of bounds, so the ball became dead at that instant. The fumble recovery didn't count; the Patriots retained possession. To come back to our theoretical case: if the Team B player fumbled, and the ball rolled into the end zone and was recovered by Team A, that's a touchback, because it was Team B that lost possession. Team A would get the ball at its own 20.

If you have possession of the ball at your own 1 and fumble it out of the back of your own endzone it is ruled a safety. If you have possession of the ball at the other team's 1 and fumble it out of the endzone it is a touchback (a touchback is awarded to defense for the offense playing it at all out of the endzone) and goes to the other team.

While it appears much thought went into the first answer, the second answer is correct.

Agreed. The second answer is right. If a team has established possession in the field of play and causes the ball to go dead through its own end zone, it's the same as any other situation where the offense kills the ball in its own end zone. It's a safety.