Disclosure: I work in diabetes research.

That "five types" article is very poor. They simply applied k-means clustering to a large set of people with diabetes. The problem with k-means clustering is that it can generally find anything you want it to. For instance, you tell k-means clustering how many clusters you want it to find, not the other way around. In this example, there is significant overlap/fuzziness between the classes. It's better to say that there is a wide spectrum of the progression of diabetes.

It is well-known that type 1 diabetes that arises late in life tends to progress very much more slowly than type 1 diabetes that arises in teenagerhood. A teenager with type 1 diabetes can have a very rapid progression, and have very severe symptoms. A person who develops type 1 in their seventies can have such a slow onset that the treating doctor can get very confused about what is going, and assume that it is type 2 instead, until the patient finally cannot live without insulin. K-means clustering has separated these two situations, but they are a spectrum from aggressive to gentle.

But the distinction between type 1 and type 2 is still very relevant. With type 1, the problem is too little insulin (progressing to zero insulin produced because the immune system destroys the insulin-producing pancreatic beta cells), and the treatment is insulin. There is no other treatment.

With type 2, the fundamental problem is too much insulin, which the body naturally produces to correct other fundamental problems, namely high blood sugar levels, which has a tight interaction with diet and obesity. The insulin encourages fat/muscle cells to suck sugar out of the blood to reduce it, but this causes other problems. A high level of insulin is then needed to keep the energy stored in the fat cells and stop it coming back into the blood as sugar, and the insulin-producing cells can eventually burn out from over-use. The treatment is to deal with the high blood sugar levels and obesity - the insulin problem is a second-order effect. Injecting more insulin into a patient with type 2 diabetes can make the problem worse - it might sort out the high blood sugar level, but it does that by cramming all that sugar into fat cells, so insulin injections are generally only used when the insulin-producing cells start failing. Diet, exercise, and blood sugar lowering drugs are better treatments. Different people respond differently to this disease, with some people storing fat around their legs (which is a more healthy option) and others storing it around their internal organs (which is really bad). K-means clustering will have separated this continuous spectrum of people into separate clusters too.

There are multiple other rarer types of diabetes that tend to be overlooked, and probably don't have much participation in the k-means clustering because of their rarity. These include the CFRD you mention, but also gestational diabetes, diabetes associated with various other more syndromic conditions (like Down syndrome), and monogenic diabetes (a single-gene defect causes an inability to correctly produce insulin - we know of double-figures types of this, some of which can be very easily and cheaply treated, and others need to be treated like type 1). There's also diabetes that is caused by surgery, if a patient has their pancreas removed.