Lexico and Imperial College London say that the difference between abbreviations and contractions is that contractions omit letters in the middle and not in the end, hence Dr, for example, is a contraction and not an abbreviation.
Contractions are a type of abbreviation in which letters from the middle of the word are omitted. Examples include: Dr (Doctor), St (Saint), Ltd (Limited), Revd (Reverend).
An abbreviation omits letters from the end of a word and a contraction omits letters from the middle of a word. <...> Contracted titles such as Dr, Mr and Mrs should not be followed by a full stop.
Larry Trask in his article on the website of University of Sussex, on the other hand, have another view, and considers Dr to be an abbreviation and not a contraction:
Abbreviations must be clearly distinguished from contractions. The key difference is that an abbreviation does not normally have a distinctive pronunciation of its own. So, for example, the abbreviation Dr is pronounced just like Doctor, the abbreviation oz is pronounced just like ounce(s) and the abbreviation e.g. is pronounced just like for example. (True, there are a few people who actually say "ee-jee" for the last one, but this practice is decidedly unusual.) A contraction, in contrast, does have its own distinctive pronunciation: for example, the contraction can't is pronounced differently from cannot, and the contraction she's is pronounced differently from she is or she has.