Instead of drawing precise lines, the magic wand selects

[Sceptic]

a wider area along an absolutely unmistakable black/white border ???

[LeviFiction]

1) Check Feather - There should be no feather, at most one pixel but there should be no feather.
2) Check Tolerance - If there is any pixel variation that you're not detecting. Such as being 1 value up or down from pure white or pure black PSP will pick that up. Make sure you're using a small enough tolerance to pick only what you actually want.
3) Check selection mode - Is the selection mode set to RGB? If not it could be selecting pixels based on something else.

[Sceptic]

Thank you very much - the "Feather" was it. I had never noticed this variable. Maybe, it wasn't available in my previous version x3?