Words are nominative units. They are used as names for things, people, ideas, activities, etc.

That's why all nominative units are 'proper' and 'real'.

There are different types of words: simple words, compounds, abbreviations, idioms.

As the diversity of words is very wide and confusable l prefer the term 'lexical units'.

As about abbreviated forms, we should distinguish 'lexical abbreviations' (bike, bus, phone, Doc, etc.) and 'graphic abbreviations' (Mr., Mrs., Dr., etc.).

Lexical abbreviations are 'shortened words'. So they are 'proper' words. Graphic abbreviations are symbols. They are not pronounced as words and are used for easier writing and reading.

In that other discussion, the nature of a proper word was exemplified with "can I use them in my essays?"

This is important, because a word being 'proper' or not, that is, acceptable in general usage, depends on the context. Much to my ongoing annoyance, lol has now become a word accepted in spoken informal English.

So, if we make the context formal writing, we know that some abbreviations are acceptable, and some aren't. Some aren't used much outside of formal writing, such as et al. or Q.E.D., and some can be used anywhere, like CIA. Some end up becoming accepted as words in their own right, despite starting as initialisms or abbreviations, like radar. Some are acceptable in formal writing in some subjects and not others, like technical abbreviations will be fine in computer science writing that aren't in English literature writing.

As with most questions about what is acceptable, beyond the basics, it's all about context.