I finally added an apple touch icon to the website. It's not the most exciting icon I could think of.. but hey it is definitely better than the default one. To use the new icon on iOS or Android make sure to reload the page if it was already in Safari, and then use the "Add to Home Screen" feature.

From what I gather it should also work when adding a bookmark with the default Android browser, however I could not test. Please let me know if it works or not!