Here is my interpretation. The mad-mullahs are vulnerable, because the sanctions have weakened the economy a lot. Oil price decline adds to that, and the mullahs know the Saudis are in part behind oil prices. And they know the Saudis are clamoring for nukes themselves.
The mad-mullahs have a lot to be gained by giving in on their nukes, and then lying to their people about how they prevailed with the West, and got a great deal.
So to answer the question "how come the US and Iranian mullahs give different versions?"
The mullahs are lying, to save face, to save face and keep power. This is example of why such regimes try to control information, etc.
The facts will be considered by the US Senate, and I believe my version will turn out to be accurate.
The Republicans do NOT have 2/3 to override a veto, either.