I read here yesterday that it is called “The Robert Byrd rule”. Is that incorrect?
Cloture is one name for the mechanism where a supermajority can call for ending debate and taking the vote. Without cloture, debate is supposed to continue, not end. A properly function deliberative body disposes of an issue after taking it up. Disposal is either passing or rejecting, or a vote to formally set it aside, "table" it.
The Senate used to have, and most deliberative bodies have "I move we vote." The Senate got rid of that in 1808 or thereabout. The Senate still ran reasonably well, disposing of what it took up, with a vote.
Around 1917, Treaty of Versailles, IIRC, a minority in the Senate refused to vote, and nothing could move the body closer to voting. That is when the cloture rule was introduced to the US Senate. It's a synonym for "I move we take the vote." FWIW, Roberts Rules of Order has a cloture mechanism too.
What Byrd did was change the rules so that the body could have more than one subject up for discussion at the same time. So, whatever it is that the senate want so debate without talking about, could be set aside, and a new subject could be brought up so the Senators could look like they were deliberating something. The item awaiting a vote on a cloture motion is still the business of the Senate, and so is the new subject.
As a deliberative body, the US Senate is by far the most dysfunctional one the world has ever seen. They can take comfort in the fact that the public is nearly totally ignorant and easily fooled.