Docs/Examples: Remove remaining legacy auth references and align workflow examples with AuthSessionName/AuthSessionBroker terminology

[blindzero]

Problem Statement

The repository has already moved toward the current provider/auth model, but some documentation and examples still appear to contain older auth terminology or outdated request-path usage.

This creates avoidable confusion because users now have to distinguish between the intended model and legacy-looking examples. In particular, workflow examples should consistently reflect the current naming and object model:

• provider auth broker configuration belongs to Providers.AuthSessionBroker
• step/runtime selection belongs to With.AuthSessionName
• optional runtime tuning belongs to With.AuthSessionOptions

Any remaining examples using older naming or old request paths make it harder for users to understand the supported pattern.

Proposed Solution

Perform a focused repository-wide cleanup for provider/auth naming and related examples.

Scope:

• Review docs, examples, and inline sample content for outdated auth naming.
• Replace remaining legacy-looking references with the current terminology:
  • AuthSessionBroker
  • AuthSessionName
  • AuthSessionOptions
• Review workflow template examples for outdated auth request paths and update them to the current supported model.
• Ensure provider docs and workflow examples use the same wording and structure.
• Add or update tests where example validation exists.

This should be treated as a consistency and completeness issue, not as a redesign.

Alternatives Considered

Alternative A: Leave mixed terminology in place and rely on user interpretation

This was rejected because it increases cognitive load and makes the documentation less trustworthy.

Alternative B: Add compatibility notes everywhere old wording appears

This was rejected because the goal should be one consistent current model, not parallel terminology.

Alternative C: Defer cleanup until a larger documentation rewrite

This was rejected because the inconsistency already affects current examples and can mislead users today.

Impact

• Does this affect existing workflows?

  • Documentation-only and example-only changes should not affect working user workflows.
  • If example validation is introduced or tightened, outdated examples in the repository may need to be updated.

• Any backward compatibility concerns?

  • No runtime compatibility concerns are expected if this remains limited to docs/examples.
  • The only visible change is clearer terminology and more deterministic examples.

Additional Context

Suggested acceptance criteria:

1. Repository-wide search no longer finds outdated auth terminology in user-facing docs/examples where the current model should be shown.
2. Provider docs consistently describe broker configuration and runtime selection using the current names.
3. Workflow examples use current request/context access patterns and do not reference obsolete auth path styles.
4. Examples remain syntax-correct and internally consistent.
5. If repository checks exist for examples/docs, they pass after the cleanup.

Suggested review targets:

• provider documentation
• workflow template examples
• quickstart/reference/how-to examples
• inline example snippets in markdown files