Clarify Vector Database project guidance

[kosabogi]

Summary

• Add the Vector Database project type and exact value to cross-project search tag guidance.
• Clarify the Elasticsearch and Vector Database project choices in the Search getting-started flow.
• Distinguish Elasticsearch Serverless API profiles from the dedicated Vector Database project type using tabs.
• Recommend the Vector Database project type for embedding-driven workloads on the vector search landing page.

This draft follows the project positioning introduced in #7961. Open product questions are listed in a separate PR comment.

[kosabogi]

Questions to confirm

@alexmarhaba @dustincoates Before finalizing the content, could you please confirm the following?

1. Project selection: Is this the intended recommendation: use Vector Database when embeddings and similarity search are central to the workload, and use the Elasticsearch project type for general-purpose or mixed lexical, time series, and analytics workloads?
2. Agent Builder: I can see Agent Builder in a Vector Database project in QA, and the current Kibana configuration also enables it and includes it in the Vector Database navigation. However, PR #7961 currently tells users to choose the Elasticsearch project type when they need Agent Builder. Is that guidance outdated? Should Agent Builder be removed from that Elasticsearch-specific recommendation, and should we document a Vector Database-specific access path?

[github-actions]

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🔍 Preview links for changed docs

• explore-analyze/cross-project-search/cross-project-search-tags.md
• solutions/search/get-started.md
• solutions/search/vector.md
• solutions/search/vector/dense-vector.md

[github-actions]

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