For many, many years, I have been volunteering at the Mesa Regional Family History Center. During all that time, the Center has almost always been called the Mesa Family History Center. Technically, the "real" name of the Center was the Mesa, Arizona Large Multistake Center. I have always wondered why when FamilySearch had such a nice logo and name, they didn't change the names of the local centers so there was a consistent branding.
Some time ago, FamilySearch opened the Riverton FamilySearch Library and I guessed that at some point, they would see the obvious advantage of having a uniform brand name and change the names of the centers. Well, they finally did. Now the Mesa Center, as one of the 15 regional centers, is called the Mesa FamilySearch Library. Local smaller centers around the world will be called FamilySearch Centers.
There is a subtle, but important distinction in using the term "FamilySearch" rather than "Family History." You go to the centers to search for your history, not pick it up like you would in a store. You cannot believe how many people come into the Mesa FamilySearch Library and ask to "see their genealogy." There was an earlier shift in terms when FamilySearch tried to change the connotation of the term "genealogy" by associating it with the idea of "family history." As an -ology term, genealogy is more formal and speaks to the academic community. While, "family history" is something more personal and accessible.
The real question is whether or not the name changes will make any real difference in the greater genealogical community? I think the change to uniform branding of the FamilySearch centers is very positive. It gives a more direct connection with the online FamilySearch.org website and adds more exposure to the name. It also coveys the meaning change I mentioned above.
Organizationally, there is some question as to the supervision of the centers, since they are usually administered by a local Church committee, rather than being directly supervised by FamilySearch. I doubt that the name change will affect the organizational structure or their administration at all.
Meanwhile, I have to retrain myself to say "Mesa FamilySearch Library" when referring to the center. That will take some time and effort, you know old dog, new tricks and all that.