Assuming it is a fantasy world, white ink can come from the fabled Ghost Squid, if you like the look. It is a good idea to think about how the mapmaker would have done this or that, but you don't need to restrict yourself to historical possibilities if you're not doing a historical map.

Where is the lake by Badger's Drift getting its water? You might want to add some rivers coming down from the mountains to feed the lake.