docs: make code examples run and fix factual errors

[lesnik512]

Summary

Every Python block in docs/**/*.md and README.md (167 blocks) was extracted, compiled, and run in page order in a venv with the package and its extras. The blocks below failed or contradicted the source, and are fixed here. The blocks that still fail are intentional: examples that show an error being raised, placeholders (my_project, def foo(...), <YieldType>), migration-guide snippets for the old API, and the provider-overriding example that needs Postgres via testcontainers.

Changes

• providers/context-resources.md: the quick-start container raised DefaultScopeNotDefinedError (that_depends/meta.py:34-40). Added default_scope = ContextScopes.ANY. container_context(preserve_global_context=False) and add_middleware(DIContextMiddleware) with no context items raise ValueError (context_resources.py:549). Gave the first one a container and removed the second. DIContextMiddleware has no preserve_global_context (context_resources.py:767-772). The sync-context error is RuntimeError: Context is not set. Use container_context (from running the example).
• introduction/scopes.md: ContextScope.APP changed to ContextScopes.APP, return p to return val, and "providers" to "provides". The nested-scope example re-entered p in REQUEST, which raises InvalidContextError as the block above it says. Added the default_scope caveat.
• integrations/fastapi.md: the middleware example passed MyContainer and no scope. It now passes MyScopedContainer, scope=ContextScopes.REQUEST, and the test in the doc passes, opening and closing the session per request. override_context is async (providers/base.py:309), so the tip uses override_context_sync/override_sync. Also fixed container_context() with no args, Container changed to MyContainer, and a missing fetch_context_item import.
• providers/factories.md: print(msg()) and await ...provider() (factories.py:31). ContextScopes is not exported from that_depends.providers. provider_sync exists only on factories (factories.py:49), so the context example resolves a Factory. Dropped the closing sentence.
• introduction/injection.md: override_providers_sync (container.py:180) and override_context_sync. Fixed typos.
• introduction/generator-injection.md (_injected changed to injected), introduction/ioc-container.md (two SyntaxErrors), providers/object.md (object_provider() returned a coroutine, so the assert failed), index.md (logger was undefined), experimental/lazy.md (typing.Iterator[int]).
• introduction/multiple-containers.md, integrations/faststream.md, integrations/litestar.md: inlined the definitions instead of importing the repo's tests package. The Litestar lifespan is typed Litestar, not fastapi.FastAPI. The Litestar test now passes: it asserted "async resource" against a datetime resource. The FastStream example ran against TestRabbitBroker.
• README feature list, providers/resources.md intro, testing/provider-overriding.md intro, dev/contributing.md (git clone, "open to"): wording.

Checklist

• Lint and format pass (ruff)
• Type check passes (mypy and pyrefly)
• Tests pass and new behavior is covered
• Build succeeds (uv build) if packaging or build config changed
• Docs updated if behavior or public API changed
• Repo metadata stays consistent across the three surfaces (GitHub description, pyproject description, profile blurb) if this touches packaging

mkdocs build --strict passes.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.

Flag	Coverage Δ
unittests	100.00% <ø> (ø)

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