As a guy who grew up near here.....
When York region was carved up a few decades ago, the two southern-most cities, Vaughan and Markham, contained a number of communities. Markham had Thornhill (the east side), Unionville, and Markham, (and many others) that were gathered into one city. Vauaghan had Thornhill (the west side) Woodbridge, Kleinburg and Concord.
All of these "villages" continue to use their names as postal addresses. Even today, the residents consider themselves as part of that community.
Vaughan Mills is in Concord. Concord residents will claim it as theirs, but outsiders will not understand it. There are no signs saying "welcome to Concord" going to the area. They will see Vaughan signs.
For clarity of visitors, it makes better sense to call it Vaughan.
Thanks
Murray - formerly of Thornhill (Markham side)