If Tuesday's handshake between Putin and Iranian President Mahmoud Ahmadinejad troubles Mr. Bush, he didn't let it show at his news conference.
"I think it's hard to judge how their conversations went from a picture," said President Bush.
Mr. Bush is betting Putin will be on the same page opposing Iran's nuclear weapons ambitions that he was on this summer, adds Axelrod.
"We spent a lot of time talking about the Iranian issue, and we both agree," said President Bush on July 2.
In light of current problems with Russia, CBS News chief Washington correspondent Bob Schieffer noted how Mr. Bush didn't want to say anything about Putin until they talked in private about his recent trip to Iran.
"I think it was very interesting he wouldn't criticize him at all," said Schieffer.
Mr. Bush also urged the Democratic-controlled Congress to drop efforts to pass a House resolution labeling as genocide the World War I-era killing of up to 1.5 million Armenians in the final years of the Ottoman Empire.
With all the pressing responsibilities facing the nation, "One thing Congress should not be doing is sorting out the historical record of the Ottoman Empire," he said.
"Congress has little to show for all the time that has gone by" since Democrats gained control in January of both the House and the Senate, Mr. Bush said.
The president also said that Congress needs to act on mortgage relief for homeowners hit by the housing crisis, trade deals that would strengthen allies, legislation expanding U.S. markets and aid to military veterans.
Mr. Bush also said that he opposes Turkey's desire to launch a military offensive against Kurdish rebels in Iraq.
"We are making it very clear to Turkey that we don't think it is in their interest to send troops into Iraq," Mr. Bush said shortly before Turkey's parliament approved a cross-border military offensive against the Kurdish rebels in northern Iraq.
"Actually they have troops already stationed in Iraq and they've had troops stationed there for quite a while," he said. "We don't think it's in their interest to send more troops in."
Mr. Bush said he talked about Turkey with Ryan Crocker, U.S. ambassador to Iraq, and Gen. David Petraeus, the top U.S. commander in Iraq, on Wednesday. He also noted that Tariq al-Hashimi, one of Iraq's vice presidents, was in Istanbul expressing that Iraq shares Turkey's concerns about terrorist activities. But Mr. Bush said there's a better way to deal with the issue than having Turkey send additional troops into the country.
"What I'm telling you is that there's a lot of dialogue going on and that's positive," he said.
Mr. Bush used the news conference - his first since Sept. 20 - to prod the Democratic-controlled Congress to approve spending, education and health bills.
It came just a day before the House will try to override his veto of a bill expanding a popular children's health program.
"It's unlikely that veto override will succeed," Mr. Bush said noting that Democratic leaders knew that "when they sent me the bill."
Mr. Bush also defended his decision to attend a ceremony in the Rotunda of the U.S. Capitol later Wednesday to award Congress' highest civilian honor to the Dalai Lama, the exiled spiritual head of Tibet's Buddhists.
"One, I admire the Dalai Lama a lot; two, I support religious freedom, he support religious freedom," he said.
The reception the Dalai Lama was getting in the United States infuriated the Chinese government, which reviles the Buddhist leader.
On other subjects, Mr. Bush:
Mr. Bush promised that the conference in Annapolis would be "serious and substantive." He said one reason for holding the conference is to get what he called "Arab buy-in" for a Palestinian state.
"Part of the issue in the past has been that the Arab nations stood on the sidelines," Mr. Bush said. "And when a state was in reach, they weren't a part of the process, encouraging the parties to move forward."