docs: Standardize language, wording, and API links across all pages (#697)

## Summary

- Standardize opening paragraphs to follow "what it is + why it matters"
pattern
- Unify example intro phrasing to "The following example shows how
to..."
- Migrate all API reference links from raw markdown to `<ApiLink>`
component
- Trim verbose "benefits" paragraphs in pagination and streaming pages
- Add `:::tip` admonitions to convenience methods and pagination pages

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Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>