Originally posted by LS1 GTO:
"Stopped" at the same time, in the situation OP notes as they were both "stopped" at the same time to allow cross-traffic, means the vehicle turning left yields.
If the OP arrived at the stop sign first, and does not need to wait for cross-traffic, he had the right of way.
The optimal words here (and in the state laws noted it) are, "stopped at same time." Not "stopping at the same time."

In this case, both the OP and other driver were stopped at the same time to yield to the cross-traffic. Just as if they were both stopped at the same time to wait for a traffic light.
Nuances of words which could be better defined in the situation which the OP was faced with.