[DOCS] Improve reroute processor documentation structure and clarity

[leemthompo]

Closes elastic/docs-content#3364

Add detailed explanation of document processing after rerouting, including:

• How documents are processed through the new target's ingest pipeline
• Support for chaining multiple reroutes across pipelines
• Automatic cycle detection and failure handling

Reorganize content with clear section headings for better scannability:

• Operating modes (explicit destination vs data stream mode)
• Pipeline execution behavior
• Document processing after rerouting
• Field updates
• Permissions
• Configuration options
• Examples

Improve readability throughout with clearer language and better formatting.

Fixes feedback from issue raised by @ivosh requesting clarification on post-reroute document processing behavior.

[github-actions]

🔍 Preview links for changed docs

• docs/reference/enrich-processor/reroute-processor.md

[github-actions]

ℹ️ Important: Docs version tagging

👋 Thanks for updating the docs! Just a friendly reminder that our docs are now cumulative. This means all 9.x versions are documented on the same page and published off of the main branch, instead of creating separate pages for each minor version.

We use applies_to tags to mark version-specific features and changes.

Expand for a quick overview

When to use applies_to tags:

✅ At the page level to indicate which products/deployments the content applies to (mandatory)
✅ When features change state (e.g. preview, ga) in a specific version
✅ When availability differs across deployments and environments

What NOT to do:

❌ Don't remove or replace information that applies to an older version
❌ Don't add new information that applies to a specific version without an applies_to tag
❌ Don't forget that applies_to tags can be used at the page, section, and inline level

🤔 Need help?

• Check out the cumulative docs guidelines
• Reach out in the #docs Slack channel

[felixbarny]

LGTM, I've also added a team label so that the @elastic/es-data-management team is aware of the changes.

[elasticsearchmachine]

Pinging @elastic/es-data-management (Team:Data Management)

[elasticsearchmachine]

Pinging @elastic/core-docs (Team:Docs)