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.