What you're seeing is the difference between spring growth and the harder, thinner summer growth. Early growth (spring) is rapid, the ring is lighter colored, wider, softer and porous.
Once a stain has penetrated the early growth ring, the stain is permanently part of the layer and the only way to remove it is to remove wood by sanding, planing or inlay.
The issue here is that tree trunks and limbs are water transport mechanisms with a columnar system that extends from root to twig tip. Water based thin inks will travel through the early growth ring just as water would in a live tree.
To make a checker board like this, you will need to treat the wood with a clear oil based finish that seals the wood against this water movement and leaves the wood a natural light color and then apply the black checks over the top with black paint. You could try applying the black paint first, but there will be bleed into the early growth rings which means the lines will not be well defined.