If you try to enter "Waverley" on the BRE website, to search for a time/fare, it doesn't recognise that "Waverley" is any part of any station name anywhere...
Although my Scottish friends always refer to Waverley (not even Edinburgh Waverley), the Waverley name seems not to be recognised by some parts of "the system" at all - and it's been like that for a very long time; you've long had to search for simply Edinburgh. Given that Waverley is so well known as part of the station name (in fact as the whole of the name by many people) - whether official or not - it seems silly not to at least include "Waverley" in the name of the main Edinburgh station wherever it's officially listed.
Is the idea that it's useful for Haymarket to include "Edinburgh" in its name, so taking out the "Waverley" and saying just "Edinburgh" for the other one makes the latter more obviously the main station? If that is the logic, I don't think it really makes sense.
Very usefully, Loco2 (as was), if you try to put in "Edinburgh" as a ticket destination, offers you
"Edinburgh (Waverley, city centre)" and "Haymarket (Edinburgh, west)" - as well as Edinburgh Park and Edinburgh Gateway.
That's the only place I know that shows this information in such a useful way, linking the official name to what real people call it and to its position/importance - the only problem (if you see it as such) with it being that if you just put in "Waverley", it initially accepts that without adding an "Edinburgh" - though that change is made when it displays the results of your search.
Overseas journey planners I've used haven't recognised Waverley as a UK station name at all.