I regularly use these markers for drawing on ice, and they are the best markers I have found for this purpose. They are widely used by professional ice sculptors.
Red is by far the most visible color on ice, so it’s a pity that you don’t currently have a red marker in your range.
I tried the purple one from Your shop, and although it is visible well enough on ice, it felt like the marker was already a little dry. Could it be that these markers had been stored for too long and had started to dry out?