> in which the predicate is affirmed of the subject with some qualification,
E.g. an error with an error message
Or
> or which involves the affirmation of possibility, impossibility, necessity, or contingency.
E.g. an are you sure? dialog preceeding a save.
It has to do with the notions that are expressed by "can, should, might, must, has to" etc. These words (most of them are called "modal verbs" in English) modify a proposition in such a way that they do not mean it happened/happens/whatever, but qualify it, or indicate its possibility etc. So "he's going out now" is a proposition. "He can go out now" modifies that proposition - no longer are we affirming it, but merely affirming the possibility or permissability of it.
Nothing about "are you sure" is modal. A modal dialog is called modal because the mode of the program has changed - no longer can it accept requests to delete articles or add text or modify an avatar, but it is now in a mode when you can either say "delete this" or "don't delete this".
In particular, if you showed an inline prompt that says "are you sure? delete/cancel" but still allows you to interact with the rest of the system, it's not modal.