It seems that not all House Democratic leaders agree on how to characterize the ongoing impeachment proceedings against President Trump. First, there's House Judiciary Committee Chairman Jerry Nadler, whose view is that his committee is conducting "formal impeachment proceedings" into President Trump, he told CNN a little over a month ago.

Nadler's comments on the matter came after the Judiciary Committee said in a court filing in July that it's actively considering articles of impeachment and is seeking access to redacted materials from the Mueller report in order to decide whether to move forward with the process.

But House Speaker Nancy Pelosi has been adamant that a formal impeachment inquiry has not yet begun. On Monday, she told reporters that she was not familiar with Nadler's comments about being in formal impeachment proceedings. She said that the House was "investigating" and "litigating" in its ongoing probe into Mr. Trump's behavior.

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Nadler, on Monday, referred to his committee's actions as an "impeachment inquiry," but he did not refer to the inquiry as "formal."

"It has been an impeachment inquiry and it continues to be ... We are examining the various malfeasances of the president with the view toward possibly, the possibility, of introducing, of recommending articles of impeachment to the House. That is what an impeachment inquiry is," Nadler told reporters in the Capitol Monday. "What we're doing is very clear. It's been very clear, continues to be very clear. We're taking another step in our investigation as to whether we should recommend the articles of impeachment to the House, which you can call an impeachment investigation."

House Democratic Caucus Chairman Hakeem Jeffries said he agreed with Nadler that "we're in the midst of a Judiciary Committee investigation, which is appropriate," though he stopped short of saying that the committee had begun a formal inquiry.

"As part of that investigation, of course, as [Nadler] has said, at some point in the future. We're going to have to determine whether to recommend charges that the president has engaged in high crimes, or misdemeanors," Jeffries told reporters Tuesday.

House Majority Leader Steny Hoyer was even more blunt. When asked by reporters on Wednesday if the House was in the midst of an impeachment inquiry, Hoyer simply replied: "No." But Hoyer clarified his comments in a statement Wednesday afternoon, saying: "I thought the question was in regards to whether the full House is actively considering articles of impeachment, which we are not at this time."

"I strongly support Chairman Nadler and the Judiciary Committee Democrats as they proceed with their investigation 'to determine whether to recommend articles of impeachment to the full House,' as the resolution states," Hoyer continued.

These comments by House Democratic leadership highlight the ongoing semantic confusion among caucus members about the status of the impeachment investigation. Congressman Jamie Raskin, a top member of the Judiciary Committee, told CBS News Tuesday seemed to suggest that there is an "impeachment inquiry" or "impeachment investigation" is indeed underway, and has been for some time, but it is not yet formal.

On Thursday the Judiciary Committee is scheduled to vote on formalizing the rules of the impeachment inquiry. It will also vote on whether to install new procedures for its inquiry and on intensifying its investigation into Mr. Trump.

"It's the first time that we are formalizing the rules of the impeachment inquiry that has been going on," Raskin said in describing the vote to CBS News. "The chairman said that we were in an impeachment inquiry, and we undoubtedly are trying to determine whether or not to move forward with articles of impeachment...This would authorize and formalize it."

The new procedures would allow Nadler to designate which committee and subcommittee hearings are related to the probe, give committee counsel extra time to question witnesses and receive evidence in closed executive session.

Regardless of House leadership's opinions on impeachment, it's gaining more support within the caucus. A majority of House Democrats now support opening an impeachment inquiry, and the Congressional Progressive Caucus endorsed an impeachment investigation earlier this week.

"It is impossible to ignore the mounting evidence that President Trump has repeatedly broken the law, committed impeachable offenses, and continues to obstruct justice. The Congressional Progressive Caucus strongly endorses the House Judiciary Committee's current formal impeachment investigation and the adoption of the Resolution of Investigative Procedures," the caucus said in a statement, reflecting the idea that a formal inquiry is already underway.

Nancy Cordes, Rebecca Kaplan and Kimberly Brown contributed to this report.