Style: Design System use across more components (#6705)

## Summary

Only remaining use is in `buttonTypes.ts` which @viva-jinyi is going to
be working on to consolidate our different buttons soon.

## Changes

- **What**: Replace light/dark colors with theme aware design system
tokens.

## Review Focus

Double check the chosen colors for the components

## Screenshots

| Before | After |
| ------ | ----- |
| <img width="607" height="432" alt="image"
src="https://github.com/user-attachments/assets/6c0ee6d6-819f-40b1-b775-f8b25dd18104"
/> | <img width="646" height="488" alt="image"
src="https://github.com/user-attachments/assets/9c8532de-8ac6-4b48-9021-3fd0b3e0bc63"
/> |

┆Issue is synchronized with this [Notion
page](https://www.notion.so/PR-6705-Style-WIP-Design-System-use-across-more-components-2ab6d73d365081619115fc5f87a46341)
by [Unito](https://www.unito.io)

---------

Co-authored-by: github-actions <github-actions@github.com>

tests-ui/tests/components/dialog/content/manager/NodeConflictDialogContent.test.ts

@@ -186,14 +186,14 @@ describe('NodeConflictDialogContent', () => {
 
       // Import Failed Extensions section should show 1 package
       const importFailedSection = wrapper.findAll(
-        '.w-full.flex.flex-col.bg-neutral-200'
+        '.w-full.flex.flex-col.bg-base-background'
       )[0]
       expect(importFailedSection.text()).toContain('1')
       expect(importFailedSection.text()).toContain('Import Failed Extensions')
 
       // Conflicts section should show 3 conflicts (excluding import_failed)
       const conflictsSection = wrapper.findAll(
-        '.w-full.flex.flex-col.bg-neutral-200'
+        '.w-full.flex.flex-col.bg-base-background'
       )[1]
       expect(conflictsSection.text()).toContain('3')
       expect(conflictsSection.text()).toContain('Conflicts')