The different forms of diabetes no longer lend them selves to two simple categories. Various efforts to refine the classification have been met with controversy and complicated by evolving understanding. Here is my attempt to summarize the current thinking.
Type 1: caused by complete autoimmune destruction of the beta cells. A good practical definition is that patients require exogenous insulin in order to stay alive. That is, they will invariably develop ketoacidosis (DKA) when deprived of insulin, even in the basal state. It is important to specify the basal state, because patients with other forms of diabetes can go into DKA as well, but only in the presence of some stressor such as sepsis, MI, stroke, etc. This designation has changed little in recent decades and remains useful, though it has seen some tweaks as noted below.
Type 1b aka 1.5: These designations are no longer very useful for a variety or reasons. They originally (especially 1b) referred to a group of patients in certain ethnic groups with phenotypic characteristics of both type 1 and type 2 diabetes who seemingly transitioned from type 1 to type 2 and/or back, due to a non autoimmune mechanism: intermittent reversible severe beta cell failure due to an exaggerated form of glucotoxicity. This group has subsequently been found to be more heterogeneous than previously thought, both in terms of ethnicity and pathogenesis. To confuse things further, these terms (especially 1.5) have also been used to denote late autoimmune diabetes of adulthood (LADA), an unrelated condition. The terms were partially replaced in popular usage with ketosis prone type 2 diabetes but that too has been waning in popularity, largely abandoned. The ADA, recognizing that there are patients who develop DKA but lack antibodies, created the category of “idiopathic type 1 diabetes.” A more recently proposed category recognizes the heterogeneity in these patients (and subclassifies them accordingly) and is known as ketosis prone diabetes (see below). To confuse things a bit, KPD also incorporates patients who do not fit this phenotype, in order to encompass all diabetic patients who go into ketoacidosis apart from some severe stress. (Note: a very early designation for patients seemingly transitioning between the phenotypes of DM 1 and 2 was Flatbush diabetes).
Ketosis prone diabetes (KPD): This is a proposed designation to replace the category immediately above and adds some other mechanisms, attempting to encompass all patients who spontaneously develop DKA. It recognizes the heterogeneity of the phenotype above, specifically the fact that some forms have an autoimmune pathogenesis. Its 4 categories are based on the presence or absence of beta cell reserve and the presence or absence of autoimmunity.
Type 2: DM 2 is pretty well defined and I will not spend a great deal of time here other than to caution against defining it as any case of diabetes that does not develop DKA in the basal state. That is to say that some forms of diabetes, that don’t invariably cause DKA in the basal state, are not appropriately classified as DM 2 as will be discussed below. Although DM 2 is itself heterogeneous the patients have in common insulin resistance, gradual beta cell fatigue and the metabolic syndrome.
Type 3: Here’s where it gets even more confusing. While often a wastebasket there are some forms of diabetes that rightfully belong in this category though in current literature they have varied and sometimes quite limited degrees of acceptance. There are numerous subcategories. Here they are.
Type 3, no letter designation: This is a theoretical construct that Alzheimer disease is essentially diabetes (insulin resistance) localized to the brain and might be effectively treated with insulin sensitizing agents.
Additional categories of DM 3, designated by letter, were taken from this site:
Type 3 A refers to genetic defects in beta cells, essentially MODY. Inheritance is monogenic autosomal dominant as opposed to the polygenic inheritance of DM 2.
Type 3 B refers to severe genetically determined insulin resistance as seen in Donohue syndrome and related disorders.
Type 3 C is a more accepted category and denotes diabetes due to damage to the pancreas as a whole, eg pancreatitis, pancreatic cancer or pancreatic trauma.  . This is important because it is usually misdiagnosed as DM 2 yet has unique treatment implications.
Type 3 D is DM caused by other endocrinopathies eg Cushing’s.
Type 3 E refers to DM caused by drugs such as corticosteroids.
Type 3 F refers to DM caused by infection. In the cite referenced above congenital rubella was given as the example. Would Hep C fit in here?
Type 3 G refers to diabetes associated with unusual autoimmune diseases, eg stiff person syndrome.
Type 3 H refers to diabetes associated with Down’s syndrome.
Note: Although all the entities mentioned above under type 3 are real I could find little or no independent support in the literature for the nomenclature except for the one with no letter designation (Alzheimer disease) and type 3C.
Type 4 This is a theoretical construct based on an animal model, attempting to explain some instances of apparent DM 2 in lean adults. This may not be an important entity in man if it exists at all and might be confused with LADA.
Latent autoimmune diabetes in adults (LADA). It is sometimes been referred to as DM 1.5.
Double diabetes. You could be unlucky and have both 1 & 2. Or, in DM 1, if you treat overeating with more and more and more insulin and thereby gain of sufficient weight the characteristics of DM 2 could develop secondarily.