docs: use sign in terminology across guides and style guides (#49877)

## What kind of change does this PR introduce?

Docs update. Aligns documentation and style guides with the **Sign in /
Sign out / Sign up** platform standard.

Closes DOCS-1328. Related to
[#49874](https://github.com/supabase/supabase/pull/49874).

## What is the current behavior?

Docs style guides prefer _login_ / _log in_. Guide prose uses mixed
login and sign in wording.

## What is the new behavior?

- [WORD_LIST.md](apps/docs/WORD_LIST.md) and
[copywriting.mdx](apps/design-system/content/docs/copywriting.mdx)
document the sign in standard
- Design-system auth examples updated
- Guide prose and API reference spec descriptions updated

### Terminology

**Standard:** Use _sign in_, _sign out_, and _sign up_ as verbs. Use
_sign-in_, _sign-out_, and _sign-up_ as nouns and adjectives. Match
Studio UI labels (**Sign in**, **Sign out**, **Sign up**).

**Preserved intentionally:**

| Category | Keep as-is | Example |
| -------- | ---------- | ------- |
| Feature name | social login | `/social-login`, `features.mdx` heading,
OAuth provider section |
| URL slugs | `login` in paths | `/phone-login`, `/login-flows`,
`choosing-login-flow` |
| CLI | `supabase login` / `supabase logout` | Reference ids
`supabase-login` / `supabase-logout`; executable commands unchanged |
| SDK methods | `logout()` | Kotlin/Swift method names in API reference
titles and examples |
| Third-party UI | Provider product labels | Facebook Login, Kakao
Login, portal **Login** buttons |
| Postgres | Database terminology | login privileges, login credentials,
login via role |
| Audit/logging | Log prose | "Generates the following **log** in the
Postgres Logs" |
| Code and routes | Paths and filenames | `app/login/`, `Login.tsx`,
`demos/android-login` |
| External URLs | Third-party login pages | `dash.cloudflare.com/login`,
`console.neon.tech/login`, `vercel.com/login` |
| API identifiers | Event and field names | Audit actions
`login`/`logout`, `should_logout_user` |

## To test

- Run `pnpm lint:mdx` in `apps/docs`
- Spot-check `features.mdx`, `social-login.mdx`, and a provider guide
(e.g. Facebook, Kakao)

<!-- This is an auto-generated comment: release notes by coderabbit.ai
-->
## Summary by CodeRabbit

* **Documentation**
* Standardized authentication terminology across guides, reference
material, CLI documentation, and copywriting guidance using “sign in,”
“sign out,” and “sign up.”
* Updated authentication instructions, headings, link text, examples,
and SSO guidance for clearer, more consistent wording.
* Corrected related grammar, spelling, hyphenation, and documentation
links while preserving established product names and implementation
commands.
* **Style**
  * Refined code examples with consistent import ordering and spacing.
* **Examples**
* Updated authentication button and menu labels to “Sign in” and “Sign
out.”
<!-- end of auto-generated comment: release notes by coderabbit.ai -->

apps/design-system/content/docs/copywriting.mdx

@@ -226,6 +226,10 @@ Use a fragment with no trailing period, and prefer declarative over instructiona
 | "Handle errors"  | "View and resolve errors"        |
 | "Work with data" | "Query and update data"          |
 
+## Authentication terminology
+
+Use **Sign in**, **Sign out**, and **Sign up** for button and menu labels. Keep `login`, `logout`, and `logOut` in code, routes, URL slugs, and CLI commands only when they match existing implementation names.
+
 ## Capitalization
 
 - **Sentence case** for all UI text (buttons, labels, section headings)

apps/design-system/registry/default/example/button-as-child.tsx

@@ -4,7 +4,7 @@ import { Button } from 'ui'
 export default function ButtonAsChild() {
   return (
     <Button asChild>
-      <Link href="/login">Login</Link>
+      <Link href="/login">Sign in</Link>
     </Button>
   )
 }

apps/design-system/registry/default/example/button-with-icon.tsx

@@ -2,5 +2,5 @@ import { Mail } from 'lucide-react'
 import { Button } from 'ui'
 
 export default function ButtonWithIcon() {
-  return <Button icon={<Mail className="mr-2 h-4 w-4" />}>Login with Email</Button>
+  return <Button icon={<Mail className="mr-2 h-4 w-4" />}>Sign in with email</Button>
 }

apps/design-system/registry/default/example/dropdown-menu-demo.tsx

@@ -112,7 +112,7 @@ export default function DropdownMenuDemo() {
         <DropdownMenuSeparator />
         <DropdownMenuItem>
           <LogOut className="mr-2 h-4 w-4" />
-          <span>Log out</span>
+          <span>Sign out</span>
           <DropdownMenuShortcut>⇧⌘Q</DropdownMenuShortcut>
         </DropdownMenuItem>
       </DropdownMenuContent>