What are the damage recovery mechanics in Splatoon 2?

Sometimes you take damage but it is not enough to splat you.

I get the impression that swimming in your own ink helps remove the damage, but I don't know anything specific about damage recovery, such as under what conditions it happens or how quickly it happens when it starts.

What are the rules and explicit numerical effects of damage recovery?

Swimming in your own ink does indeed recover health more quickly. From the wiki:

After taking damage, there is a one second delay before healing will begin. Touching even a small drop of enemy ink on the ground can reset this delay. In kid form, Inklings heal at a rate of 12.5 damage per second. While submerged in ink, Inklings heal at a rate of 100 damage per second.

Keep in mind that Inklings have 100 health.