chore(studio + docs) improve project integrations (#45200)

## What kind of change does this PR introduce?

Docs update and UI copy improvement.

## What is the current behavior?

The GitHub integration working directory helper uses an inline tooltip
for examples. In the form layout, the tooltip alignment and inline code
wrapping can make the helper harder to scan.

The integration settings pages also mix markdown-rendered copy with JSX
copy, duplicate the integration cover image helper, and use slightly
different heading, spacing, and button copy patterns across GitHub,
Vercel, and AWS PrivateLink.

## What is the new behavior?

The GitHub working directory field now keeps the helper copy concise and
links to a dedicated docs section with examples for repository root and
nested `supabase/` directories.

The integration settings sections now use a more consistent JSX
structure and copy treatment across GitHub, Vercel, and AWS PrivateLink,
including shared cover imagery and matching explanatory section
hierarchy.

| Before | After |
| --- | --- |
| <img width="1838" height="696" alt="CleanShot 2026-04-24 at 14 35
32@2x"
src="https://github.com/user-attachments/assets/f9843018-fbf5-4c18-a0c3-1842b665f3fd"
/> | <img width="1842" height="700" alt="CleanShot 2026-04-24 at 14 30
21@2x-64E8726E-0B89-4C5A-8A47-CE1AA24F3D1A"
src="https://github.com/user-attachments/assets/bb017464-2b43-4dfe-94b8-a586aedea318"
/> |
| <img width="1884" height="1432" alt="CleanShot 2026-04-24 at 14 40
19@2x"
src="https://github.com/user-attachments/assets/b34e4573-cd7c-4588-83a0-0e1941019552"
/> | <img width="1844" height="1424" alt="CleanShot 2026-04-24 at 14 39
02@2x"
src="https://github.com/user-attachments/assets/4f82b01c-cb27-4f01-b2da-76b7f6e298a5"
/> |


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

* **Documentation**
* Clarified GitHub integration setup: provide a Working directory value,
how to compute it from the repo root (including "." for root), examples,
and updated “initialize” spelling.

* **Refactor**
* Simplified integration form descriptions (tooltips → inline text +
Learn more link).
* Reworked GitHub and Vercel integration content to use consistent
headings, layout, and shared image handling.
  * Reorganized integration settings rendering for clearer UI states.

* **Style**
  * Minor copy, casing, and button text/spacing adjustments.
<!-- end of auto-generated comment: release notes by coderabbit.ai -->

apps/docs/content/guides/deployment/branching/github-integration.mdx

@@ -13,13 +13,19 @@ In the Supabase Dashboard:
 2. Under **GitHub Integration**, click **Authorize GitHub**.
 3. You are redirected to a GitHub authorization page. Click **Authorize Supabase**.
 4. You are redirected back to the Integrations page. Choose a GitHub repository to connect your project to.
-5. Fill in the relative path to the Supabase directory from your repository root.
+5. Set the **Working directory** field.
 6. Configure the other options as needed to automate your GitHub connection.
 7. Click **Enable integration**.
 
+### Set the working directory
+
+The working directory is the path from your repository root to the directory that contains the `supabase/` folder. Enter `.` when `supabase/` is at the repository root.
+
+If `supabase/` is nested deeper in your repository, enter its parent directory instead. For example, if your layout is `apps/web/supabase/`, enter `apps/web`.
+
 ## Preparing your Git repository
 
-You will be using the [Supabase CLI](/docs/guides/cli) to initialise your local `./supabase` directory:
+You will be using the [Supabase CLI](/docs/guides/cli) to initialize your local `./supabase` directory:
 
 <StepHikeCompact>
     <StepHikeCompact.Step step={1}>