Magic wand also makes the pixels outside the border 50% transparent

Set the tolerance to 0 in the options bar at the top of the screen. Your screenshot doesn't show what you have it set to. Setting it to 0 will make it only select 1 exact color and you will have no antialiasing. You can adjust the number as needed.

Seems to me that you have problem with anti-aliasing. One quick way to fix this when using selection tool is to create layer mask then to use Levels from Image menu on Layer mask to remove grey values from mask.

Grey values will be between white and black color on mask, greys are transition between black and white. Later you can load selection from mask or apply layer mask.