Word for abbreviations that have become standard words

There is indeed a word and that word is...acronym.

Acronyms are specifically words formed from the initial letters of other words and pronounced as words themselves. Your examples fall into this category.

Quite often, these words aren't written in capitals: laser, for example, tends to be written in lower case, whereas Nato is often written with just a capital N.

A set of initials that doesn't create a new word, eg FBI, UN, BBC, CNN, is called an initialism.

The 'app' example is slightly different. This is an abbreviation where a word has simply been shortened. More established examples include phone, bike, fridge, gym, photo. As far as I'm aware, there's no special word for these.

(I'm writing from a British English perspective.)

When an acronym gains wide acceptance and gets listed on standard dictionaries as a word rather than a short form, e.g., laser, you know you have a word.

The ultimate test could be the appearance of inflections: laser -> lased, lasing.

Until then, it has to be content being an acronym.