Blend the two types, permanent and washable, with a powdered colorant, in the proportions appropriate to get as much hardness as possible but not so much that it is difficult to clean up the screen. The powder should be a ceramic stain mix with a melter medium (a glaze or frit).
The logo on the left was rubber-stamped using and ink mix made of only glycerine and Mason 6666 black stain. The glaze is shedding off during firing. Multiple properties needed by a stamping ink are not present here. First, the stain dries as a powder, it has no hardening or bonding properties, glycerine is its only mechanism. Second, it is too concentrated, the black color is so powerful that it bleeds excessively into the overlying glaze. Third, it does not melt during firing so it does not bond with the body below. And, it either develops only a fragile interface with the glaze above, or sheds it off. The piece on the right mixes the stain 50:50 with a glossy transparent glaze (having 20% kaolin), it lays down better, accepts the overglaze layer better (because it has less glycerine), presents less problems in handling before glazing and it has no issues with the overglaze crawling off during firing. Black stains are potent, a 75:25 stain:glaze mix would work even better.
Silk screen printing
Silk screen printing is one of the best options for hobbyists and potters to reproduce crisp and detailed decoration for their ware. But there are many details to know.
It is a mistake to use pure stains for decorating ware. Stains need to be mixed with a ceramic medium and a working medium to enable effect use and produce a good fired result.