When you look at the Area 51 proposal for Writing StackExchange, it displays as Writers. But, when you head to the actual site, it gives its name as Writing. Is this because the name has been edited since proposal? If so, why hasn't it been updated?
Area 51 is out of date. "Writers" changed to "Writing" last year.
Some of the time, the Area 51 proposal has been manually edited to reflect changes. You can see the revision history by clicking on "X years ago" under "edited" in the sidebar on Area 51 (if you don't see this, it probably hasn't ever been edited; it works just like posts). Movies is a good example of a Area 51 proposal that was edited after its expansion in name/scope to "Movies and TV". "Beer" was also edited to be "Beer, Wine, and Spirits" two years ago.
For whatever reason, the moderation team hasn't yet updated the names of many sites. "Community Building" is still "Moderators" despite that change happing many years ago. "Psychology and Neuroscience" is still "Cognitive Sciences". There's no "Veganism" in the listing for Veganism & Vegetarianism. The list goes on, even though there's really no good reason for these changes to the name not to be made. The reason is probably just that nobody thought to make these changes.
And then there's that site that's had several name changes: "Not Programming Related" to "Programmers" to now "Software Engineering". Area 51 still says "Not Programming Related". Because of the drastic scope change, the description for the proposal was edited and there's a banner mentioning all these changes. Not mentioning all this history would make proposed questions like "What's good music for programmers to listen to?" seem way out of place. I think this is the exception rather than the rule.
The Stack Overflow in <Language> proposals are also exceptions, since they are listed with an English name and a bilingual description in Area 51, something that wouldn't be good on the actual site.