An adverse drug event ADE is any occurence that happens during the course of a drug therapy that may or may not be directly due to the drug. For example, a patient on ciprofloxicin experiences vomiting once, cause unknown. An adverse drug reaction ADR is an occurence that has been shown to be related to a course of drug therapy. For example, vomiting due to gastric ulceration associated with long term NSAID use.

An ADR can be a lack of response to a drug therapy, so a patient on insulin but is not responding to that version of insulin could be classified as an ADR.

In short, all events during drug therapies are ADEs, only if they are proven to be associated with the drug do they move to ADR classification.

this indeed depends on the causality, so without additional information, this will likely classify as an ADR (and this is by definition also and ADE). Commonly the next step is also to assess severity (grading), not similar to seriousness.

It is still an adverse drug reaction because insulin has a direct causal relationship with hypoglycemia. Although, ADR are a subset of ADE. It includes things like a fall, fracture or car accident from the resulting hypoglycemia. I think your doubt comes in because the patient is still using insulin despite the hypoglycemia. Since the definition of a side effect is an unintended outcome resulting from taking the correct dosage of medication for a correct indication. Where as toxic effect is more or less the opposite. You consider the hypoglycemia no longer the correct indication. However, the patient is still diabetic, so insulin is still indicated. However, more information is required. Such as missing meals, renal dysfunctional (may require lower insulin dose), autonomic dysfunction, does the patient have insulin dependent DM, etc. For now it is an ADR until you provide information to show otherwise.