What is a casserole (vessel)?

In the US casserole is a general term for a slowly baked dish with multiple ingrediants. I would call a lasagne a type of casserole.

The typical dish is a baking dish.

Handles are not required but common. Useful for other slowly baked items such as meat.

This is from a UK perspective. A casserole dish is generally round or oval, almost always has a lid, and can go in the oven. It may be made of a variety of materials.

The use that earns the name is cooking dishes consisting of ingredients cut up and in a liquid sauce/gravy, without a topping except possibly dumplings. The terms casserole and stew have a lot in common. The dishes may be used for other things as well.

Some can also be used on top of the stove (metal ones, a few special ceramic examples). This is handy for browning ingredients before adding the liquid. There's obvious overlap with a (fairly large) saucepan, but unlike casserole dishes saucepans may have long handles, which might not be suitable for oven use either because of size or material.

A milk pan (not pot) is a small saucepan with a long handle, usually with a lid, and sometimes with a spout. In a set of saucepans it would be the smallest.