I think there’s considerable debate among historians about what signal was used to signify death or mercy.
But you are correct about how very inaccurate Hollywood’s portrayal of ancient Rome has generally been.
In case we get the idea that The Games weren’t as bloody as you might think, what with most of the contests only to first blood, they had “special events.”
Theatrical enactments of myths, with actual live rape, dismemberment, burning alive on stage, etc.
At intermission criminals were fed to animals. The criminals sometimes included Christians. On one memorable occasion under the Republic, a wealthy aristo was murdered. The law required all his slaves to be executed for not preventing it. So several hundred men, women and children were executed at the Games.
Nero wrapped Christians in shrouds soaked in pitch, mounted them on poles and used them for lighting.
Captured enemy soldiers were often forced to fight each other to the death two at a time, with the winner fighting the next in line, up to a hundred or more. The last man standing was generally freed.
For many centuries, the rare criticism of the Games was based not on their brutality, which the Romans considered positive, but on their popularity with the lower classes.
Historians think over 500,000 people were killed in the Coliseum during its four-five hundred year run.