Watching Barack Obama work his charismatic magic on Channel 2 interviewer Yonit Levy on Thursday night, and through her on the Israeli public, one was quickly reminded afresh of how it was that this remarkable politician defied immensely improbable odds to become president.

The impressionable Levy may not have been too difficult to win over, but the Israeli mainstream – battered, bloodied and instinctively cynical these days about peace prospects – is a rather tougher nut to crack, particularly when the message is that familiar one about narrow windows of opportunity, Palestinian willingness to make concessions, and the need to overcome fear in order to achieve change and lasting security.

But Obama – who carefully dropped seemingly casual references to the Jewish concept of “tikkun” and to his visits to the Western Wall into the conversation – likely took at least a partial step toward reeling in our skeptical public with a performance that was focused, well-prepared and engaging.

This was his first Israeli TV interview since he won the presidency, and his first interview with the Israeli media since he sat down with this writer in Jerusalem as mere Democratic frontrunner Obama in July 2008.

And while I was struck, in that conversation two years ago, by what I wrote then was his “explicit and unsympathetic” attitude to the matter of West Bank settlements – he told me that Israel would have to consider whether “getting that buffer [of an expanded Israel] is worth the antagonism of the other party” – it seems equally significant that he was so vague and non-confrontational when the same issue was raised by Levy.

The president told her he didn’t want to be “disingenuous” and to claim that there were no differences between him and Prime Minister Binyamin Netanyahu. But, he went on amiably, the US and Israel had always differed over settlements, and the US position was formulated only in the belief that it would help ensure a strong, secure Israel. He easily dodged her direct question about extending the 10-month settlement moratorium, and steered gently away instead to declare, unarguably, that if only the direct talks could get started, they would hopefully produce the kind of growing Israeli-Palestinian trust that would enable all kinds of knotty issues to be addressed more productively.

The president ticked all the right boxes – stressing his “sympathy and identity” with the Jewish experience; disarmingly acknowledging that his middle name “Hussein” might prompt suspicion, then offsetting that by naming his senior Jewish advisers; and noting that thwarting Iran’s nuclear drive had been his “No. 1 foreign policy priority.” He also astutely praised Netanyahu as a leader “not perceived as a dove” and all the more capable of peacemaking as a consequence.

The only slightly sour moment was his musing over whether some Israelis’ wariness of him had been caused by his outreach to the Muslim world, and the cliché he invoked about the critics who wrongly believe that “the friend of my enemy must be my enemy.”

For in truth, of course, we Israelis would like to think that we and he share precisely the same friends and precisely the same enemies, and our concern has sometimes been that he was simply not as experienced as we are at separating the one from the other.

Perhaps most importantly of all in this masterly performance, the president, when urging a more flexible attitude to peacemaking from the Israeli public, did so with a commendable effort at empathy: “The Israeli people are going to have to overcome legitimate skepticism,” he said, and “more than legitimate fears,” in order to achieve the breakthrough that would enable long-term security.

“Legitimate skepticism.” “More than legitimate fears.” One could imagine Israelis nationwide nodding their heads at that language, beginning to concede that this leader does actually understand something of what we’ve been going through.

Two years ago, candidate Obama and I had discussed his pledge to work for an Israeli-Palestinian accommodation from the moment he was sworn in, if he was sworn in. Obviously, it is unfortunate he didn’t conduct the kind of interview he gave to Channel 2 right at the start of his presidency.

But it’s a safe bet that a watching Israel was overwhelmingly gratified that he has done so now.