I did find this fascinating article about dip pens, but it doesn't quite answer the question although it does seem to confirm my thought that travelling / pocket inkwells were more about being carried from place to place rather than pulled out and used throughout the day:
https://oro.open.ac.uk/84894/3/84894.pdf
It strikes me that a pencil would have been far more convenient for tasks such as jotting in a notebook, and that anyone expecting you to sign and write in ink would also have ink at hand for the purpose. Indelible pencils did not appear until the later 1800s, so they would not be an option.