Generalize the remaining base-cli context and configuration API

[codeforester]

Summary

Generalize the remaining public context and configuration API after the Base adapter extraction.

The lifecycle is consumer-neutral, but public names such as base_home, UserConfig, and workspace-specific fields still reflect the original Base consumer. Replace or deprecate those names with consumer-neutral contracts without forcing standalone applications to understand Base terminology.

Scope

• define neutral application-home and user-configuration types or callback values;
• migrate Base to the neutral API through its explicit profile;
• retain narrowly scoped compatibility aliases only where they are needed for one migration release;
• update docs, type checks, and tests;
• keep package rename out of scope.

Dependencies

• follows the adapter extraction tracked by Move legacy Base adapters out of base-cli #22.

Acceptance criteria

• generic lifecycle and public types contain no Base-specific field names;
• Base-specific aliases are documented as temporary and are not required by standalone consumers;
• Base and base-cli test suites cover the migration and compatibility behavior.