Bill Tindall, E.Tn.
The ink is made by dispersing pigments in something. The "something" will dissolve but the pigments are highly insoluble. As Jordan said, the only way to be rid of them is scrubbing and given their color intensity even if you scrub away most you will remain colored until that layer of skin falls off.

The YouTube recipe is a testimony to the failure of high school chemistry classes to teach chemistry. Other than the abrasive in the toothpaste there is nothing in that mix that would be useful for this task, or any other I can think of. Any time someone recommends mixing an acid (vinegar and lemon juice, AKA acetic acid and citric acid) and a base (baking soda , AKA sodium bicarbonate) to make some even more effective product than either alone one can be assured the source of this suggestion failed chemistry.