In days of analog scopes it was always prudent to turn down the intensity.
Before embedded computers it was possible to burn the phosphor off the inside of the screen by powering up with sweep sped set slow and intensity high.
There's a concept named "Writing Speed"
in the time it takes dot to go across the screen, how many electrons have hit the screen ? Enough to hurt it ?
Remember current is Coulombs per second.
At sweep speed of 1 msec/cm, a second's worth of beam current illuminates 10 meters worth of screen
At sweep speed of 1 sec/cm, a second's worth of beam current illuminates only 1 cm of screen. The phosphors don't get to cool off between sweeps.
That's writing speed, how many cm per second of trace are to be lit. Obviously low writing speed requires less current in the electron beam that makes the dot..
So we were taught to form the habit of setting intensity knob all the way down when shutting off the scope.
Many Tektronix scopes had their power switch on the intensity knob like a radio volume control, forcing you to think about writing speed when you turned the scope on..