Will means "it will happen" and Would means "it wouldn't happen"?

Question

Compare will and would:

• I'll stay a little longer. I've got plenty of time.

I'd stay a little longer, but I really have to go now. (so I can't stay longer)

• I'll call Lisa. I have her number.

I'd call Lisa, but I don't have her number. (so I can't call her)

(Taken from English grammar book)

From what I've understood after reading the examples above, can I conclude that will is used to express desire in the future spontaneously (my book told me to use "be going to" when the event is planned), but wouldn't is used to say something that failed to happen (wouldn't happen)? Is that correct?

By the way, these are some references that I've read, but got no explicit information about what I'm asking.

https://english.stackexchange.com/questions/3657/when-should-i-use-would-would-have-will-and-will-have

https://english.stackexchange.com/questions/155239/difference-between-would-and-will

https://english.stackexchange.com/questions/67931/correct-use-of-will-would

Most explanations said would is just the past form of will or in another case, would is softer than will when it's used to ask someone to do a favor or to offer something to someone. (Would you like..., would you please..., etc).

In my case, there's no unambiguous clause using the past verbs, so why do we use "would"?

To make sure that I have understood, here I made some sentences and let me know whether they're correct or not. Thanks.

1. I'll buy some books tonight.

I'd buy some books, but I don't have money. (I won't be able to buy some books)

2. I'll paint the wall because there's a stain on it.

I'd paint the wall, but I don't have paint. (I won't be able to paint the wall)

Answer 1

Use of modal verbs is one of the most subtle aspects of English grammar, and it is not helped at all that usage is in rapid flux.

Historically, will/would, can/could, may/might, and shall/should were present and past forms. However, “will” and “shall” designated futurity, intention, or obligation so talking about “present” tense is a bit misleading.

He will go

is talking about an event that is expected or intended to occur in the future, not one that is occurring in the present.

Moreover, the “past” forms are often used in conditional sentences that have nothing to with past time.

If I knew her number, I would call her.

Structurally, the verbs are in the past tense, but the meaning is present. We are discussing events that are contrary to fact at the present time.

The examples you are talking about reflect differences in fact.

I will stay; I have plenty of time.

Future intention at present.

I would stay if I had time.

Counter-factual in present.

I cannot stay because I have no time.

Factual and present form discussing future reality.

As I said, usage of modal verbs is complex, and I am not trying to be exhaustive, but the use of “will” and “would” to distinguish between what is implied to be true and what is admitted to be false is common and important.