It took three hours and 33,000-plus words, including 57 mentions of “jobs,” seven “Bibi’s” and one “Big Bird.” But the first two debates revealed something that long months of speechifying had not: two campaigns, stripped down to their core arguments.

On Tuesday night, President Obama and Mitt Romney will take the stage again, in a town-hall style debate at Hofstra University in New York. This meeting may have a different tone from their first — Obama, in particular, is under pressure to be more aggressive. But both candidates seem likely to return to the same basic sales pitch.

Romney and his running mate, Rep. Paul Ryan (Wis.), have focused largely on their policy agenda, which was dominated by things they wanted to roll back — the health-care law, the Dodd-Frank financial regulations, income-tax rates.

The Democrats tend to cast the election less as a choice between ideas and more as a choice between people. Both Obama and Vice President Biden stressed their personal credibility, citing their work in office and the lessons of their upbringing. They attacked Romney and Ryan as aloof and uncaring.

“Look, folks, use your common sense,” Biden said Thursday night — talking specifically about Social Security, but also compressing the Democratic appeal into two sentences. “Who do you trust on this?”

The importance of the campaign’s first two debates was made clear in recent days by the zeal with which Obama and Romeny prepared for the third. They sequestered themselves in swing-state hotels — Romney in Ohio, Obama in Williamsburg — engaging in mock debates at the expense of real-life campaigning.

It was worth it, the campaigns calculated, because of the powerful impact the first debate had on the race. The slow and steady gains that Obama had consolidated last month were all but erased by his poor showing.

The vice-presidential debate, as usual, had less of an impact. Both sides were pleased with the performances by Biden and Ryan, but the sole vice-presidential debate does not appear to have fundamentally changed the race.

The final debate will be held Oct. 22 in Florida and will focus on foreign policy.

Still, even if the first debates left the race itself muddled, they served to clarify the messages at each campaign’s heart.

That’s because, for once, there were time limits. Some issues had to be left out. None of the candidates, for instance, has mentioned climate change, immigration or same-sex marriage.

And — just four years after Obama built his candidacy around a promise to change Washington’s partisan culture — this year, the two nominees had to be prompted into talking about it at all. Romney said he’d sit down with Democrats on Day One. But Obama noted that Romney also wants to start rolling back the health-care law that day.

So, Obama implied, Democrats would already be mad.

The arguments the two sides did put forward, in their limited time onstage, signaled sharply different theories of what voters want.

Romney and Ryan seemed to imagine the American electorate like a business trying to hire a contractor.

In that view, voters would want to see a work plan first, even if some details were left negotiable. Ryan seemed to mock Obama for not understanding that and leaving out details of his plan to reduce the deficit.

“That’s what we get in this administration: speeches,” Ryan said. “But we’re not getting leadership.”

In the debates, the two Republicans did spend some time talking about Romney’s business experience and generosity. Ryan, for instance, told a story about Romney helping members of his church, whose two sons had been left paralyzed by an auto accident.

“Mitt Romney’s a good man. He cares about 100 percent of Americans in this country,” Ryan said.

But the two Republicans spent far more time talking about their ideas to strengthen the country’s military, and to pare back other parts of government.

In many cases, the two still left the details of these ideas fuzzy. Romney said he wanted replacement bills on health care and financial regulation but spelled out only a few specifics. He also declined to specify how he would cut tax breaks, to make sure that tax revenue doesn’t decline when overall rates are cut.

“I come in and lay down a piece of legislation and say, ‘It’s my way or the highway,’ I don’t get a lot done,” Romney said.

The Democrats had a different idea about what voters wanted: Obama and Biden seemed to imagine the public like a football team trying to hire a coach. So they would care about personality, and experience, more than any specific list of plays.

It was a decision that could, and should, be made with one’s gut.

“Folks, follow your instincts on this one,” Biden said during his debate, talking specifically about Medicare.

Both Democrats dwelled on one specific policy idea: the proposed “Buffett Rule,” which would raise tax rates on some high earners.

But not many others. Obama, in fact, lapsed into a familiar campaign-trail habit: phrasing his second-term agenda in terms of questions, not answers. “How do we deal with our tax code? And how do we make sure that we are reducing spending in a responsible way?” he said in the debate with Romney.

To make the argument that Democrats should be trusted to answer questions like these, Obama and Biden cited their efforts to bail out General Motors, the imposition of sanctions against Iran and the successful mission to kill Osama bin Laden.

Biden stressed his long experience in foreign-policy matters — especially after Ryan referred to Israeli Prime Minister Benjamin Netanyahu by the familiar nickname “Bibi.”

In Biden’s response, he used the same nickname three times in two sentences.

“Now, with regard to Bibi — who’s been my friend 39 years — the president has met with Bibi a dozen times,” Biden said. “He’s spoken to Bibi Netanyahu as much as he’s spoken to anybody.” Three more “Bibi’s” followed before the end of the debate.

Biden also attacked Romney as untrustworthy, citing the secretly videotaped speech in which Romney said 47 percent of the U.S. population consider themselves “victims.”

“He’s talking about my mother and father. He’s talking about the places I grew up in,” Biden said.

So what would Biden and Obama do for those people?

True to form, Biden didn’t mention a specific bill or even an idea for a bill. Instead, he described the noble — and highly intangible — goals that he and Obama shared.

“The president and I are not going to rest until that playing field is leveled, they, in fact, have a clear shot and they have peace of mind,” Biden said. “Until they can turn to their kid and say with a degree of confidence: ‘Honey, it’s going to be okay. It’s going to be okay.’ ”