Bashar al-Assad had barely settled into his new quarters in Russia before the argument broke out over who can take the credit for ousting him, ending 53 years and two generations of brutal family rule over Syria.

President Biden and his aides say they set the stage, because they worked relentlessly to weaken Syria’s main backers, including Russia, Iran and Hezbollah. There was a reason, Mr. Biden argued, that none of Mr. al-Assad’s allies were able or willing to come bail him out at the very moment he needed rescue.

Prime Minister Benjamin Netanyahu of Israel and his aides tell a very different story, saying the only reason Mr. al-Assad fell was that Israel killed the head of Hezbollah, Hassan Nasrallah, which Mr. Netanyahu called “a turning point in the collapse of the axis.” Israeli officials have been quick to note that Mr. Netanyahu ordered that attack over the objections of the Biden administration, which feared that going after Mr. Nasrallah would risk a wider war.

And Mr. Netanyahu inserted the knife a bit himself, saying that the key was his decision to ignore pressure “to stop the war before we accomplished all of our goals.” His message was clear: Had he listened to Mr. Biden’s warnings about avoiding a “wider war,’’ Mr. al-Assad would most likely still be lounging in his palace this week.

Sorting out the truth here is not easy, and historians of the Middle East will most likely be arguing for years over what factors were at play, not least the Syrian rebel force itself, which had crucial support from Turkey. Certainly, if there was an American contribution to Mr. al-Assad’s fall, it took a while: In August 2011, President Barack Obama said it was time for Mr. al-Assad to “step aside,” and two years later, on his first visit to Israel as president, he stood with Mr. Netanyahu and declared that Mr. al-Assad had lost all legitimacy and “must go.”

But there was no real plan to oust him, and when it finally happened last Sunday, the unraveling of the regime came so fast that American intelligence officials were taken by surprise about the speed of the collapse.

Suzanne Maloney, a scholar at the Brookings Institution who closely follows Iran and the region, noted on Tuesday that Mr. Biden may be right to take credit for creating some pressure, but she said there clearly was no American plan to oust the al-Assad government.

“I think it is probably true that U.S. policy contributed to the fall of Assad,” she said, “in that the U.S. has helped weaken Iran’s regional reach, and kept Russia from maintaining the tempo of its military support for Assad.”

But she added, “It is probably untoward for the Biden administration to take credit — it was ultimately Syrians who took the actions that freed them from this brutal regime.”

“There was no evidence of a wider U.S. strategy that was aimed at the outcome that we have seen in Syria over the past week,” she concluded. “Washington took its eye off the ball in Syria many years ago.”

For Mr. Biden, the collapse in Syria came just as he is giving a series of talks meant to cement his legacy in many areas, from the steady growth of the American economy during his time in office to the re-establishment of alliances. And Mr. Netanyahu, facing a corruption trial at home and hard questions about why his government did not see the Oct. 7, 2023, terror attack by Hamas coming, is out to rescue his reputation.

So claiming credit for the toppling of a regime that has been the central funnel of Iranian arms, money and influence — and that used chemical weapons against its own people in a long civil war — is pretty irresistible for two politicians at the tail end of long careers. And, to add to the stirring of the pot, neither man makes much effort to hide how much he dislikes the other.

The differences of approach continued on Tuesday. Israel attacked what was left of Syria’s Navy and hit sites associated with Syria’s chemical weapons program. The Israeli defense minister, Israel Katz, discussed how the military “has been operating in Syria in recent days to hit and destroy strategic capabilities” that pose a particular risk. At the White House, John Kirby, a spokesman for the National Security Council, neither condoned nor criticized the targeting of these sites, and he would not say if the United States was providing the Israelis with intelligence that would aid the operation. “We have robust intelligence sharing with Israel, as you might expect,” he said. But he declined to say “what that looks like” in the current operations.

But Mr. Kirby did make a case that if Russia had successfully taken over Ukraine, or if Iran had succeeded in its recent missile attacks on Israel, both countries would have been in a better position to aid their longtime Syrian ally. Instead, they stayed out of the fight. Mr. Kirby said that was no accident.

“I don’t think that anybody should overemphasize one particular operation, whether it’s against Hamas or against Hezbollah, as being the game-changer,” he added. But he said Mr. Biden’s move, after the presidential election, to reach a cease-fire with Hezbollah sent a serious signal. The rebel groups “paid attention to the fact that Hezbollah was no longer in the fight.”

In Mr. Netanyahu’s telling of events, there would have been no American-brokered cease-fire if there had been no Israeli attack on Hezbollah, starting with the explosion of beepers and cellphones that killed some members of the terror group’s leadership — and deeply unnerved its fighters. Then came the destruction of much of Hezbollah’s stockpile of missiles and the killing of its leadership.

In short, Mr. Netanyahu is arguing that had he listened to Mr. Biden, Hezbollah would still be in power in Lebanon and Mr. al-Assad would still be in Damascus. Instead, Mr. al-Assad’s palaces are being looted.

All of this raises the question of how President-elect Donald J. Trump will alter the approach, if at all. Just before Mr. al-Assad fled the country, Mr. Trump posted on Truth Social that the American strategy should be to sit on its hands.

“Syria is a mess, but is not our friend, & THE UNITED STATES SHOULD HAVE NOTHING TO DO WITH IT,” he wrote. “THIS IS NOT OUR FIGHT. LET IT PLAY OUT. DO NOT GET INVOLVED!”

That approach is very much in line with Mr. Trump’s avowed adherence to an “America First” philosophy. But in his first term, Mr. Trump was talked out of his instinct to withdraw the small American troop presence in eastern Syria that fights ISIS. And the American interest in a peaceful transfer of power, a freely elected government and the elimination of chemical and biological stockpiles in Syria is obvious.

So far, Mr. Trump has not tried to reconcile his all-caps warning with the stakes for the United States in making sure Syria is not another power vacuum for terrorists, jihadists or anyone looking to use those chemical and biological weapons. And, as some experts note, there is a clear intersection between Mr. Trump’s stated desire to set back Iran and his interests in Syria’s future.

“There is a unique opportunity to weaken Iran and build the pro-Western coalition that President Joe Biden has been seeking,” Itamar Rabinovich, a former Israeli ambassador to the United States and once Israel’s chief negotiator with Syria, wrote on Monday. “But his administration is on its way out and President-elect Donald Trump has posted on social media that the United States should have 'nothing to do’ with the Syrian ‘mess.’”

He argued that “it is key that his advisers make yet another effort to persuade him that Washington lead and win the current struggle for Syria.”