The ongoing debate over confirmation of former Sen. Chuck Hagel to be Defense secretary took a detour Thursday into a partisan disagreement over what constitutes a filibuster.

Majority Leader Harry Reid and Senate Democrats said the 58-40 cloture vote, short of the 60 needed to limit debate, represented a filibuster by GOP senators.

Republicans “have made an unfortunate choice to ratchet up the level of obstruction here in Washington,” the Nevada Democrat said. “Just when you thought things couldn’t get worse, it gets worse.”

Supporters of more robust changes to the Senate’s rules than were adopted in January — meant to reduce the number of filibusters — said the Hagel situation could push Reid to do more to address the procedure.

“I think the important thing is that the leader is getting concerned,” said Sen. Tom Udall, D-N.M., a leader of the effort to make bigger changes to the rules regarding the filibuster. He added that he interpreted Reid’s comments as signaling that “if this is going to continue, he’s going to look at other ways to make the institution more productive and less bogged down.”

Republicans, meanwhile, dismissed the notion that the cloture vote amounted to a filibustering of Hagel, who even GOP senators admit is likely to get the votes needed for confirmation after the weeklong Presidents Day recess.

“It’s not a filibuster,” Sen. Johnny Isakson, R-Ga., said. “All we’re doing is extending debate. We could have another vote tomorrow.”

The question is whether a cloture motion being filed necessarily has a cause-and-effect relationship with an objection being present. In some cases, Reid has filed such motions pre-emptively. In this instance, however, several GOP senators made it clear that Hagel had not provided the information they sought to move forward.

“I think we’d like for the vote to happen … after recess so there’s enough time for some of the senators who’ve asked legitimate questions — some have gone over the top, but the ones who’ve asked legitimate questions — to have them answered,” Sen. Bob Corker, R-Tenn., said. “This is not like blocking; this is like saying we don’t want to end debate yet.”

In an online fireside chat, President Barack Obama scolded Republicans for holding up Hagel’s nomination, saying a filibuster of a Defense secretary is unprecedented.

“We don’t have a 60-vote rule,” Obama said.

Democrats also tried to argue that it’s dangerous for the Pentagon to be leaderless while Hagel’s nomination is in limbo.

“During times like this, it’s nice to have a secretary of Defense. Not a lame duck,” Reid said.

Pentagon spokesman George Little said Thursday that outgoing Defense Secretary Leon E. Panetta will leave his post after Hagel, a former GOP senator from Nebraska, is confirmed, “which the secretary hopes happens as quickly as possible.”

Sen. John McCain, R-Ariz., alluded to that, saying no one was forcing Panetta to leave.

“We do have an obligation of advise and consent — a constitutional responsibility — and we’re trying to carry this out,” McCain said.

Sen. Lamar Alexander, R-Tenn., has personal experience as a Cabinet nominee subject to a delay. Then-Sen. Howard M. Metzenbaum, D-Ohio, held up Alexander’s nomination to be President George Bush’s Education secretary for about three months.

“I think it’s nothing more than an example of the majority leader moving prematurely to end debate,” Alexander said. “In the case of the secretary of Defense, you would think that you allow more that two days after a nomination comes out of committee to allow senators to ask questions and consider the record.”

While senators could not even agree on how to define a filibuster, some of the arguing sounded like a bid to confuse the umpires.

Sarah Binder, a scholar at George Washington University and the Brookings Institution who studies the Senate’s rules, wrote Thursday on the Monkey Cage blog that the action on Hagel should rightly be called a filibuster using a “duck test”: “If it looks like a duck, swims like a duck and quacks like a duck, then … it’s a filibuster!”

The Senate rules changes adopted in January would, among other changes, cut down on the amount of debate required after cloture has already been invoked on nominations short of the Cabinet level and create an expedited process for considering lower-level judicial nominations. None of the changes related to nominations eliminate the ability of a senator to force a procedural vote with a 60-vote threshold to cut off a filibuster.

Other changes give Reid more flexibility in moving legislation to the floor, but those modifications also fundamentally preserve the supermajority requirements. George Kohl, senior director of the Communications Workers of America, one of the groups that led the push from the outside for rules changes, said Reid should go further.

“Reid should keep the Senate in session and force the debate to continue to explain why Republicans are intent on blocking a fellow Republican nominee and decorated war veteran,” Kohl said. “Leader Reid accepted the obligation to sustain debate under these terms when he agreed to continue unchanged filibuster rules.”

But McCain, who helped draft the rules agreement, said he didn’t think the Hagel matter was related to the package approved by the Senate earlier in the session to reduce the number of filibusters and allow more amendments to legislation.

“I don’t see it, frankly, as a test,” he said. “Our rules were that if you do filibuster … that you better stay there for your hour and line up others.”

John M. Donnelly and Steven T. Dennis contributed to this report.