coddled eggs

You can also bake them in the oven! Break into the (buttered) coddlers, add a little cream, and maybe a little seasoning, put on the tops and place in a pan of hot water, bake gently 6-10 mins. in a pre-heated 350º oven, and call them shirred eggs.

A true Coddled egg is cooked in an egg coddler, usually porcelain, with a screw top metal lid. Submersed in boiling water, it very gently cooks the egg, as the egg is coddled by the coddler. I grew up in Bermuda with my very British and proper Grandmother, who would have had a heart attack if you ate the egg directly from the coddler!
The coddler was used in substition for poached eggs, it was more foolproof and never wasted any of the egg, and required less attention. Once cooked, a quick run with a pairing knife around the egg, as you would to release a custard cup, and you have a perfectly poached, coddled egg, nicely shaped. She had both single and double coddlers, I still enjoy them today, and yes sometimes add a tablespoon or two of cream to make it a Cocotte, french style.

Yes, to me there are two types.
The English use nice porcelain cups (with lids) and coddle the egg as described above the eat with a spoon and dip dry toast points in it (because the yolk is warm and runny)...
Or,
As I do for my Caesar salads, I simply hold the egg under hot running water go a minute (and over the last 40 years NObody has been ill).

A coddled egg is a gently cooked egg. It's similar to a poached egg, but it's cooked in its shell or the egg coddler that you mentioned. They can be barely cooked as for a Caesar salad (I just use raw eggs from healthy pastured hens), a little more cooked which is fun for ramen or to top bibimbap all the way to soft boiled stage, but I always think of them less cooked than soft boiled.