What is a casserole (vessel)? Maybe it's because I'm German speaking, but I'm utterly confused by the concept of the concept of a Casserole.

When I enter "Kasserolle" in the German Amazon I on the one side get dishes that look like that they could get put into the oven. But on the other side I get pots which have a long grip. In my family we used to call them "milk pots". 

Are these milk pots also supposed to be put into the oven? I've never seen anybody put pots like these into the oven, but doesn't the word "casserole" usually imply recipes that require an oven? I am utterly confused by the concept of "casserole". 

Why does it seem to imply two different kinds of dishes ("milk pots" and dishes I'd use to make a lasagne)?
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.