The most common name I've seen for this is "emptying the enemy's guns" or similar. For example, from Grinell (1915), pg 74:

One of the Cheyenne chiefs called out to his people; "These people want to fight; now let us get ready and kill them." And Medicine Water answered, saying: "My son, Alights on the Cloud, will empty their guns."

Then, when everything was ready. Alights on the Cloud rode twice around the Delawares and close to them, and they all shot at him, emptying their guns as they tried to kill him, but the shots did not harm him. While they were shooting at him the brave Sioux on foot almost reached the end of the Delaware line. Now the Cheyennes all made a charge and the Delawares, having nothing in their guns, ran back and down into the ravine, where their horses were, but before they had time to load again the Cheyennes were upon them and killed them all. Some of them had the ramrods in their guns, the balls only half-way down the barrels.

Another example can be seen in Vestal : https://books.google.com/books?id=pnneBQAAQBAJ&pg=PA114&lpg=PA114

One reason for this "trick" to be done by riding across the enemy line at close range is to take advantage of the 1/20 to 1/10 second delay between pulling the trigger and the bullet leaving the gun with a flintlock musket. The time delay is due to the powder in the priming pan burning first, then the powder in the touch hole, and finally the main gunpowder charge that propels the bullet. This delay is in addition to the reaction time of the shooter. With a percussion musket, or breech-loading cartridge rifle, one only has the reaction time of the shooter, and the tactic is more dangerous.

Less dangerous, but less effective, are other enticements to shoot. For example, from the testimony of Presley Talbot, a company commander at Sand Creek, at the Sand Creek Massacre hearings:

Firing ceased for not more than five minutes; one Indian, which proved to be Big Head, who as a signal showed buffalo robe to the height of a person, as the means of drawing the fire from the soldiers, so that they would empty their guns, and then would give a whoop and rise en masse and fire arrows, shot muskets, and squirrel rifles.

With the adoption of repeating rifles, this tactic ceased to be effective for the purpose of creating a window of opportunity for a charge while the enemy was reloading. Deliberately drawing fire could still be used to exhaust the enemy's ammunition.

References

George Bird Grinnell, The Fighting Cheyennes, C. Scribner's Sons, 1915. https://archive.org/details/fightingcheyenn00gringoog

United States Congress, Senate. Report of the Secretary of War, Sand Creek Massacre, Sen. Exec. Doc. No. 26, 39 Cong., 2 sess. Washington, Government Printing Office, 1867. Relevant extract at http://www.kclonewolf.com/History/SandCreek/sc-documents/sc-05hearing.html

Stanley Vestal, Sitting Bull: Champion of the Sioux, University of Oklahoma Press, 2014 (originally published 1932). https://books.google.com/books?id=pnneBQAAQBAJ