It was likely Reza lying about his insiders in Iran and Bibi going along with it to Trump because he wants an excuse to be pardoned from his trial(war)
I doubt this.
For one, Netanyahu would like to stay out of prison but this isn't the cause of this war. Polls showed the overwhelming majority of Israelis want this war. Opposition leader Lapid and Bennet have shown support for it. Its part of the Greater Israel vision to either balkanize Iran or replace it with a friendly regime.
As for Reza, I do not believe for a moment Trump or any American official believed his grift. Trump publicly dismissed Pahlavi multiple times, and the guy has no cohesive movement or support within Iran. He's less viable than Venezuela's opposition leader, and we saw what she got: Nothing.
I think it's more to do with elections and I am not talking about US midterm elections.
Iran proxy war through Hamaz and Hezbollah is not hidden from anyone. There's strong dislike for Iran in Israel. You can say popular sentiments is in favor of taking action against Iran.
Bibi has been advocating actions against Iran for a long time as well but his hands were tied.
This year Israel has legislative elections and survey was showing that Bibi's party was behind in polls.
I think war against Iran became a domestic political necessity for Bibi.
At the same time, Trump was on hot seat due to Greenland fiasco followed by Epstein files.
So perhaps both thought a campaign against Iran would help them more in elections.
But I don't think they expected Khamenei's death and Iran's hard stance which followed leading to Hormuz control.
Now, you can clearly see the difference in their stance.
Bibi says it was bound to a long term campaign while Trump who isn't so diplomatically tuned said it was weeks long campaign.
Short campaign is also evident from build up around Hormuz before war started. I doubt US can make such a strategic blunder if they were planning or were anticipating long campaign.
Like I said above: the Iran war has massive public support in Israel.
And this Iran war started last year in June, it became clear Israel couldn't finish the job on its own and this is why America is fighting the war now with Israel.
But you're right on a very important point: seems like Israel and the USA have different objectives for this war. Israel wants to bring down the regime and that includes destroying infrastructure to cause an uprising, while the USA wants a longer and slower degradation of Iran's capabilities while keeping infrastructure for whoever they think will rule Iran next.
As for Greenlannd, I think Trump would be more serious about taking it now than before since NATO has shown it has less desire to take part in America's adventures and would rather keep the defensive alliance a purely defensive one, especially after the Ukraine war.