I am trying to boil your post down. It seems as though your immediate supervisor fired you, but that decision was overruled by his boss. Is that correct?
If so, then you quit and weren't fired.
Since this is going on a background investigation, a written clarification may be needed. Check with your background investigator. Where the issue may lie is when they contact that employer and what HR says.
My background investigation didn't care whether I was fired or quit. They wanted to know if I ever left or was released from a job under unfavorable circumstances. They wanted to catch the stories where someone quit before they could be fired.
In time we hate that which we often fear.
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