TypeScript SDK: coordinator schema-mapping tests (camelCase to snake_case)

[jason810496]

Background

The public TypeScript options are camelCase while the supervisor schema uses snake_case fields. Tests validating this serialization mapping were deferred in the SDK public-interface PR.

What needs to happen

1. Add coordinator schema-mapping tests that validate camelCase public options serialize to the supervisor schema's snake_case fields.

Acceptance criteria

• Every public option is covered by a test asserting its serialized snake_case counterpart.

Context

• Add TypeScript task SDK public interface #67908 (PR description)