Why is the word raptor not written as 'raptor when used as the abbreviated name for velociraptor?
There are two things going on here. The first is that raptor is not cleanly synonymous in this sense with velociraptor because the word is used as both a synonym and a hypernym; raptor being also used as an informal name for dromæosauridæ generally as they all share the bird-like features (both confirmed and suspected) that raptor evokes (due to its further senses of birds of prey) and because several sub-groups, genera and species have names ending in -raptor.* As such even when using raptor as a synonym of velociraptor there is an introduced vagueness because the word you are using is also a hypernym. This means it is not as clean a synonym as with some other cases where one may choose between an informal and scientific name.
Note that while in many of these cases we can arrive at raptor by contraction, we cannot do so from the word that covers them all; dromæosauridæ. Raptor in this sense is not a contraction, but an expansion of a word formed by contraction or finding of common elements to cover a wider meaning.
Another more general point about the English language is that we don't tend to think of such coinages as contractions as consciously as we once did.
Up until around a century ago, it would have been very common to use apostrophes to indicate elision, and so we have don't, fo'c's'le, 'twas and so on.
Then it became more common to just abbreviate by contraction or elision without marking where the elision had been made, and so 'bus competed with bus, 'phone competed with phone and so on, depending on the styles of different writers and editors.
Since then the apostrophe has lost more and more ground in this use (and lost some other uses entirely or all but entirely). It is still used in this case, particularly when the writer fears the use might be obscure otherwise or is transcribing speech patterns they don't use themselves†, but we are much quicker to treat such contractions as independent words than as contractions requiring apostrophes, than once we were.
*There are luanchuanraptor, pamparaptor, variraptor, pyroraptor (these last two may be the same species), austroraptor, buitreraptor, neuquenraptor, tianyuraptor, graciliraptor, changyuraptor, microraptor, (these last four all examples of microraptorinæ), bambiraptor, atrociraptor, linheraptor (which like velociraptor is an example of velociraptorinæ) & utahraptor. There are also some which do not share this -raptor ending.
†I'd be more likely to writer 'king 'puter than king puter not just to avoid king becoming ambiguous bewteen a contraction of fucking and a male monarch, but also because that's not how I personally would swear at a computer.