[3.9] Fixed the style of the search-boxes in the dark theme and the reDocs pages

[vamera]

Description

This PR fixes the style of the search boxes in the dark-theme and some bugs in the style of the reDocs pages.

Checks

Docs building

• [x] Compiles without warnings.

  Code formatting and web optimization

• [ ] Uses three spaces indentation.
• [ ] Adds or updates meta descriptions accordingly.
• [ ] Updates the redirects.js script if necessary (check this guide).

  Writing style

• [ ] Uses present tense, active voice, and semi-formal registry.
• [ ] Uses short, simple sentences.
• [ ] Uses bold for user interface elements, italics for key terms or emphasis, and code font for Bash commands, file names, REST paths, and code.