docs: improve database instance connection documentation (#833)

- Restructured documentation with clearer navigation and organization
- Added network requirements section at the top for better troubleshooting
- Simplified docker commands to show only relevant environment variables
- Improved section headers and navigation with Quick Start cards
- Consolidated secret manager options in General Setup section
- Maintained all original screenshots and detailed instructions
- Fixed terminology (database instances instead of databases)
- Removed redundant troubleshooting section
- Enhanced readability with better formatting and structure

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Co-authored-by: Claude <noreply@anthropic.com>

Author: d-bytebase