Leaders of so-called rebel republics in Donetsk and Luhansk held referendums in May 2014 that declared independence.
It was the first clarification that had to be issued after a number of deeply controversial comments made by Mr Trump during that news conference.
Never mind that the White House has spent three days trying to clean up the political fallout from the summit amid bipartisan criticism, or that the special counsel investigation into Russian meddling in 2016 continues apace.
Republicans and Democrats both accused him of siding with an adversary rather than his own country after he shied away from criticizing the Russian leader for what USA intelligence agencies say were Moscow's efforts to undermine the 2016 election.
In a group of tweets, Trump said that his meeting with Putin was "a great success", and he looks forward to their second meeting so as to "start implementing some of the many things discussed", including stopping terrorism, security for Israel, nuclear proliferation, cyber attacks, trade, Ukraine, Middle East peace and more.
But, he threatened a downturn in their relationship could be bad news for Russian Federation.
She said the US hopes Mr Putin will have the Russians indicted on charges brought by the US Department of Justice "come to the United States to prove their innocence or guilt".
Director of National Intelligence Dan Coats was also unaware that Putin was being invited to Washington.
Earlier on Thursday, the USA president had said of Mr Putin: "I look forward to our second meeting".
He said he is "far tougher on Russian Federation than any President in many years, many years".
"Gingrich added that the Trump administration is now getting into its next phase of foreign policy, including with China, North Korea and Russian Federation, and he predicts the American side will be 'remarkably tough".
Trump's comments were characterized as "disgraceful" and "shameful" by US leaders from both parties.
White House spokeswoman Sarah Sanders said that meeting may come this fall. There is US$380 million approved in the current budget for the programme, which is meant to help states strengthen election systems from hacking and other cyberattacks.
But, Coats added, Trump was right to point out this week that Russian interference is just one of many potential threats facing the country.
Just like I consider myself to be responsible for things that happen in this country.
"And so, make no mistake about it, we are very clear with our adversaries, when it's in our best interest, to share with them what we're doing", he said.
Last week, following Trump's widely panned summit with the Russian president, Coats offered a wary assessment at odds with the president's desire for a closer relationship with the former head of the KGB. He restated the US intelligence assessment about Russian meddling and Moscow's "ongoing, pervasive efforts to undermine our democracy".
Instead, standing next to Putin Trump cast doubt on the agencies.
He also lashed out at Europe and USA -dominated North Atlantic Treaty Organisation, saying Russian Federation would hit back with an "equivalent response" to the alliance's bases near Russia's borders and other "aggressive steps".