Your test page prints fine at this end. No ‘change of colour’ in the cell(s).
In the right column, the jpg with a black background prints with that background and obviously removing the black from within Word removes any black border around the perforations.
Using a different printer might give a clue to any printer issues/settings.

In the meantime though, try this as a test:
Using the test document you have supplied, delete the 3 triangular stamps and file extension letters.
Check that the table shading has 15% grey
Insert a stamp image into the top cell (use a PNG like the 'smaller zipped .png' with transparent background you attached earlier)
Right click the image
Format Picture
Fill & Line > Fill > Solid Fill > Color > 15% Grey
Repeat for other cells and check Print Preview/Print out.
Click the image below to see a short video of the above.



I appreciate it’s the printing out of the document that shows the colour change (or not).