The words are called modal verbs:
The modal verbs of English are a small class of auxiliary verbs used mostly to express modality (properties such as possibility, obligation, etc.).
The principal English modal verbs are can, could, may, might, must, shall, should, will and would. Certain other verbs are sometimes, but not always, classed as modals; these include ought, had better, and (in certain uses) dare and need. Verbs which share only some of the characteristics of the principal modals are sometimes called "semimodals".Modal verbs
Would not and did not means the same in your example.
Here are examples with the list of words you gave and explanations as to why we would or would not choose them.
- The car couldn't start this morning, so I was late for work.
The car can normally start by itself, but this morning it couldn't. We are not quite there yet.
- The car shouldn't start this morning, so I was late for work.
I disable the car so it cannot not drive off without me, but I forgot this morning and it left without me. We are also not there yet
- The car wouldn't start this morning, so I was late for work.
This is the correct word. The
would is used to give some kind of human behaviour to your car. It is as if the car chose to not start just to spite you, but it is what we normally use when we do not use didn't.
We cannot use hadn't start - it should be hadn't started to work in this context.