It seems clear from Star Trek III: The Search for Spock that the method of killing someone is normally up to the discretion of the Klingon ordering or carrying out the execution.
In many situation, Klingons in authority have life-and-death authority over their subordinates. In Star Trek III, the Klingon commander Kruge (played—very much against type—by Christopher Lloyd) first executes his agent and love interest Valkris, because she knows too much about the Genesis project.
Then, even more brazenly, he kills his gunner for no other reasons than being extremely displeased with his performance and to make the gunner's failure an object example to the rest of his crew. Kruge carries out the killing with his personal sidearm, which he then points his second officer, Torg, threatening to kill him too, if Torg criticizes what Kruge has just done.
The fact that the captain has total power to kill those working under him, on the spur of the moment, suggests that there is not likely to be a standardized procedure for conducting all executions. It would seem to be against the spirit of giving Klingon officers absolute power over their subordinates to require them to conduct executions according to some prescribed methods. The killing is done at the commander's discretion, according to whatever means he deems appropriate.
Later in the film, Kruge orders another execution—of one of the three hostages that his marines are holding on the Genesis planet. Kruge seems not to care about how the prisoner is executed. In fact, he says that it does not even matter which of the three his killed. The Klingons still down on the planet choose to carry out the execute with a barbed dagger, rather than a phaser. Once again, absent specific instructions, the way the execution in to be carried out is left to the killer's discretion.
