According to a number of media outlets, the primary message that President Obama delivered to Senate Democrats during their private meeting yesterday was that they should not “draw lines in the sand” in the standoff with Republicans over spending, the deficit, and the debt ceiling. That would seem to mean that the President isn’t committed to drawing a firm line against Medicare cuts, as liberals want him to.

But leading liberal Senator Sherrod Brown tells me he took a very different message away from the meeting. He left convinced that Obama knows it would be folly to give ground on Medicare — and that the President understands that the 2012 election will be about jobs, not the deficit.

Brown says that he told the President in the meeting that a pivot to jobs will be essential as the reelection campaign heats up, and that Obama agreed.

“The president totally understands that the 2010 election — Republicans say it was about spending, but it was about jobs,” Brown said, describing the message he took from the meeting. “He said that jobs are the most important thing in the next year and a half. He understands that.”

Brown declined to elaborate on what the President said about how he’ll pivot more sharply to jobs in coming months, and declined to share specifics about what Obama said on Medicare. But Brown indicated that the President signaled that he thinks the way to win on Medicare is to keep the contrast sharp with Republicans, in order to make it clear to the public that the GOP is governing from the extreme ideological right.

“We’re going to make the contrast on Medicare,” Brown said, characterizing the discussion. “Republicans are going to lose this debate, because the country is not on their side. Republicans are governing by ideology.”

Asked if he left the meeting with any sense that the President wasn’t fully committed to drawing a hard line on Medicare, Brown said: “I’m not worried.”