I'm looking for the English name for this cooking utensil:  It can be used for roasting on top of the stove, or it can be put in the oven for slow cooking. I've stumbled upon 'roasting dish', or 'baking dish'. But when I search for these terms, I get results that differ from what I'm actually looking for:   Source: Google image search 
To me, that's a Dutch (*) oven. You could also call it a "heavy braising pot." Often, they are enameled, so you could call it "enameled pot" also. In general, a dish doesn't have a lid, and a pot does, so a roasting or baking dish will generally be that open thing in your lower pictures. (The exception that proves the rules will be casserole dishes, some of which do have lids, but may not be ok for the stove.) (*) - not because it's from the Netherlands, but because it lets you do baking-like things without an oven. Same as Dutch treat, Dutch courage, and Dutch uncle - in the US, it means not-really. (The UK does something similar with French.) In an outdoor cooking situation, a cast iron dutch oven that sits on a fire and has a small twiggy fire actually set burning on its lid will give you quite oven-like results for some foods. Le Creuset, probably the most famous brand of the enameled version, calls them French Ovens and Dutch Ovens interchangeably on their Canadian website. I have no connection with the firm. 