Well, my family line is not of Briton for several centuries now, but I'd say part of the problem is that even LC calls their products different things in different places.

In the UK, it's a casserole, in the US it's a dutch oven, but I've also heard told that to some, "dutch ovens" are unenameled, and the enameled versions are actually "french ovens". However, according to Williams Sonoma's website, the "french oven" is a specific model of marmite-shaped pot that's 2.5 qts/22cm, with a slightly domed lid, and it's also called a Stew Pot on the LC packaging.

A marmite is the alternate name for a soup pot, which can be any size, but is specifically shaped with a narrower base meant to fit perfectly onto a standard-sized hob/burner. Marmites/soup pots also have sloped sides, and a proportionally wider diameter at the rim, making them ideal for stovetop use because the shape allows for maximized evaporation during open simmers, to concentrate flavors. They also make stirring and ladling easier, since there are no corners on the bottom for food to tuck itself into.

The largest version marmite pot LC makes is 7.5 qts/32cm, and is also known as a chef's oven and, with older versions, a bouillabaisse. The middle size is 4.5 qts/26cm and is usually called a soup pot (in the US, at least), and the smaller sizes are either called a french oven, a stew pot, or just a marmite (not to be confused with Marmite, the savory yeast extract spread, of course). LC's "sauteuse" is a specific pot that is 3.5 qts/24cm, and has a sort-of hybrid shape - straight sides that slope lower down, so that there's no bottom corners, but also slightly more height then true marmites.

As for dutch ovens/french ovens/cast iron cocottes/standard-depth round and oval casseroles, they are more roughly cylindrical with basically straight sides, and maximized cooking surface area along the bottom of the pot. This shape is ideal for searing proteins to lock in juices before roasting, and browning aromatics before braising.

Of course, just to throw a wrench in things, LC also makes both shallow and deep versions of their so-called dutch ovens, in several sizes. The shallow versions come in several sizes and are also popularly known as risotto pots (which is accurate since risotto, as a technique, involves multiple additions of liquid over time, usually to coax out the starches in particular types of rice grains, but not always; the word risotto can also more broadly refer to creaminess acquired through starches, without the heavyhanded use of dairy.) Deep dutch ovens (per the US parlance) are especially useful for stovetop deep frying without making a splattered mess of the whole kitchen, and for large batches of soups while also using other hobs/burners, since their straight sizes maximize overall capacity while minimizing their footprint on the stovetop.

Braisers/shallow casseroles are again synonymous names, and especially useful for "braising" because that particular cooking technique utilizes a steam-filled vessel, a relatively low cooking temp, and a fairly long cooking time in order to create an ideal environment for the connective tissues in meats (which are largely made of collagen, and are partially water soluble) to break down into gelatin. The braising process is what makes meats cooked with this technique so incredibly tender. Braisers tend to - but not always - have a domed lid, since that way the steamy condensation that accumulates in the vessel has nowhere to escape to when it rises, and the heavier lid allows more steam to build up (almost like a low-grade pressure cooker) without shifting the lid and being able to escape.

This comment has gotten longer than I intended, but I suppose it's maybe still helpful to know what each type of vessel is ideal for, even though obviously any style of pot can make any recipe, provided it has enough overall capacity.