Ain't necessarily so (other than the protection thing). While you've described iron gall ink nicely, india ink (lampblack suspended in dilute shellac) would be

Ain't necessarily so (other than the protection thing). While you've described iron gall ink nicely, india ink (lampblack suspended in dilute shellac) would be a paint under your definition, as would many printers' inks (pigments ground into an oil or polymer binder), especially those used for serigraphy. Soaking in isn't necessarily a characteristic of inks either; they are often brittle films (which is why palimpsests exist—old ink was merely scraped off of the parchment/vellum and the ground reused). By the same token, watercolours (which are pretty much universally regarded as paints) do soak into unsized papers. This is another one of those "where do you draw the line" questions where the line is rather broad and ill-defined. It's a continuum. When it comes right down to it, the intended purpose and traditional usage in the field determine what is an ink, what is a paint and what is a stain.