Update `applies_to` badge and popover

[bmorelli25]

Summary

Update the applies_to badge and popover according to new designs. @florent-leborgne has outlined all requirements in this doc (sorry, Elastic access only).

Related issues

• [Feature Request]: LLM readable applies_to #1589 (LLM MD implementation)
• Support applies_to in additional directives/components #1436 (custom directive implementation)
• Implement multiple lifecycle state logic #1435 (removing burden of ordering from writers)

[reakaleek]

Steps:

• Add tests to prepare for refactoring: Add comprehensive applies_to tests #1873
• Refactor applies_to. Remove logic from view and move to it somewhere else. Refactor applies_to #1878
• Combine applies_to badges with the same key: Combine applies_to badges with the same key #1900
• Reuse the logic for the popover badge