[CPS] Add CPS examples

[szabosteve]

Summary

Relates to https://github.com/elastic/docs-content-internal/issues/31

This PR adds CPS examples to the CPS overview page.

Generative AI disclosure

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Tool(s) and model(s) used: gpt 5.2 in Codex

[github-actions]

Vale Linting Results

Summary: 1 warning found

⚠️ Warnings (1)

File	Line	Rule	Message
explore-analyze/cross-project-search.md	361	Elastic.QuotesPunctuation	Place punctuation inside closing quotation marks.

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[github-actions]

🔍 Preview links for changed docs

• explore-analyze/cross-project-search.md

[quux00]

Found a few things to fix.

[quux00]

nit: I think this will read as:

in different cross-project scenarios in Cross-project search

which seems a little redudant?

Maybe:

in different {{cps-init}} scenarios

[quux00]

Change include_ccs_metadata to include_execution_metadata. Sorry, I used the wrong one in my examples to you.

include_ccs_metadata is (unofficially) deprecated - we don't want to use it for two reasons:

1. it has "ccs" in it so confusing if it applies to cps
2. include_execution_metadata will include execution metadata even for queries that are origin-only, whereas include_ccs_metadata will not

[quux00]

Change include_ccs_metadata to include_execution_metadata.

[quux00]

Change include_ccs_metadata to include_execution_metadata.

[quux00]

The search runs only in projects where the my-index index exists.

That's not true for qualified expressions. When you specify *:my-index, that index (or alias or datastream) must be present on every project, otherwise an error will be thrown:

Here's a case where the "logs" index exists on the linked project but not on the origin project:

$ ▶ curl -X POST --url 'http://localhost:9200/*:logs/_search' --header 'Authorization: ApiKey essu_dev_xxxx' | jq .
{
  "error": {
    "root_cause": [
      {
        "type": "index_not_found_exception",
        "reason": "no such index [*:logs]",
        "index_uuid": "_na_",
        "index": "*:logs"
      }
    ],
    "type": "index_not_found_exception",
    "reason": "no such index [*:logs]",
    "index_uuid": "_na_",
    "index": "*:logs"
  },
  "status": 404
}