fix(search): ensure cursor in dark variant has correct colour

[nuria1110]

fix #7814

Proposed behaviour

When the Search component is rendered with the "dark" variant and no searchButton, the cursor is always white.

Screen.Recording.2026-03-18.at.11.51.43.mov

Current behaviour

When the Search component is rendered with the "dark" variant and no searchButton, the cursor is black when input is empty, which causes colour contrast checks to fail.

Screen.Recording.2026-03-18.at.11.49.51.mov

Checklist

• Commits follow our style guide
• Related issues linked in commit messages if required
• Screenshots are included in the PR if useful
• All themes are supported if required
• Unit tests added or updated if required
• Playwright automation tests added or updated if required
• Storybook added or updated if required
• Translations added or updated (including creating or amending translation keys table in storybook) if required
• Typescript d.ts file added or updated if required
• Related docs have been updated if required

QA

• Tested in provided StackBlitz sandbox/Storybook
• Add new Playwright test coverage if required
• Carbon implementation matches Design System/designs
• UI Tests GitHub check reviewed if required

Additional context

Testing instructions