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.

There are many common examples which are older than laser. The word snafu, for example, was originally an obscene U.S. Army reference (situation normal -- all f'ed up"), and fad is purportedly originally an acronym for "for a day". Some terms, such as AWOL, seem to be stuck in between-- they are almost used as words, but equally almost always capitalized. Arguably the most commonly borrowed English word, "ok" is generally considered an acronym for the slang phrase "orl korrect".

I refer to them as maws (matriculated acronystic words), which reuses an otherwise rarely used three-letter word (how often does the stomach of a ruminant come up in everyday conversation?), and makes a nice pun on English' propensity to easily digest and swallow up words, whatever their source.

I think it is interesting to look at the reverse phenomena: how an otherwise common word becomes an acronym. Creating an acronym out of a common word seems to be a favorite of computer scientists. For example, they currently use SOAP (simple object access protocols) for internet traffic. Years ago, programmers referred to large database objects as LOBs, and Binary Large Objects as BLOBs (originally all caps, but now frequently referred to in lower case as blobs). Although the term is only used in specialized circles, it really has become its word there. Legislatures also get into the act of naming to create an acronym. For example, the Revoke Excessive Policies that Encroach on American Liberties Act thus can be referred to as the REPEAL Act.

In my opinion the reverse terms are begging to be called RAWs, or Reverse Acronystic Words.

App when used as an actual word is special case of an abbreviation. As soon as you speak about it as a word of its own, it's the result of clipping. Sadly that result does not seem to have a name itself.

In German words that only retain the first part are called 'Kopfwort' (head + word), but i can't quite make up a good English equivalent right now. An "apronym" is the German word for an acronym that is constructed to hit existing words (CARE package, PATRIOT Act). The latter would at least apply to the CARE package in the OP's question.

Not to be confused with a backronym which is an acronym you make up after the fact to stand for an existing word.

Modem is not the other hand is a actually syllabic abbreviation and not an acronym, since originally an acronym would only include first letters. If you want to count it as acronym seems a matter of perspective and definition of acronym.

The rest are classic acronyms, since the word acronym isn't around that long (1940s) it might be time to create a word for words for which most people forgot - or never knew about - it being an acronym.

With the word being self-sufficient one might name it an autononym, independonym or soveronym.