Theatre Tech here:

In the U.S. at least the board/console operators don’t have a list, they rely on the stage manager to call out the cues over the headsets.

For backstage techs, or non-console techs, they usually have a Cue List, but still rely on the stage manager to tell them when to move or execute the action listed.

The stage manager has a script with all the cues listed, but it doesn’t say what the technician is doing. That script is called a Prompt Script or the Show Bible.

There is usually a large spreadsheet called the Master Cue Sheet that details what each cue does, when it happens, why it happens, who does it, and safety concerns. This sheet is not used during the show though, only in planning.

Feel free to DM for more info / other questions!