I was adding some profiles to the Western Cemetery in Cheshunt, Hertfordshire and needed to create the Category. Since the official name, as best I could tell, doesn't include the word Jewish, I didn't put it in the category name, Category:Western Cemetery, Cheshunt, Hertfordshire. I have marked it in the Affiliation parameter of the infobox.
I have now returned to the categories in this post. I agree that the existing category should become the one for the Federation of Synagogues cemetery. I am however reconsidering how to name them.
In theory, the Federation of Synagogues one does not require Jewish since many would not that Synagogue relates to Judaism. However, there is another Edmonton Cemetery, the word Jewish does help to make it clear. It also makes it easier to find when categorising a profile.
The Western one would not be correct to add Synagogue to its name. Historically, the cemeteries were affiliated with the Western Synagogue. Now, the synagogue has changed its name and the burial society uses the name Western Charitable Foundation. I am thinking to call this one Category:Western Cemetery, Edmonton, Middlesex. The same reservation about not using the word Jewish applies here too.
For clarity, all 3 cemeteries are run under orthodox Jewish lines and only allow the burial of Jewish people.