Clicking on that link takes you to a search page where you are informed as follows"
Here is another screen shot of the page:FamilySearch offers a network of more than 4500 facilities offering public access to genealogical records from all over the world. These range from the world’s largest in Salt Lake City to locations in more than 80 countries.Find a FamilySearch Center near you
It is fairly obvious to any one familiar with FamilySearch.org that the term "FamilySearch Centers" is the same as the older term, "Family History Centers," as in Mesa Regional Family History Center and Utah Valley Regional Family History Center. I would guess, as I have observed before, that there is a change in the branding. But during the interim they may be some confusion, especially because the main link on lds.org also refers to Family History Centers.
Personally, I like the term FamilySearch Centers better than Family History Center. It gives the potential user the idea that the center is a place to search not just come in and find some one has done their family history. There is always the not too unbelievable story going around of the lady who comes into the family history center, leaving her husband in the parking lot with the car running, and asks to have a copy of her "family history." I have really talked to people who thought all they had to do was come in and ask.