Refactor agent spawning and delegation check mechanisms

- Updated agent spawning from `Task()` to `Agent()` across various files to align with new standards.
- Enhanced the `code-developer` agent description to clarify its invocation context and responsibilities.
- Introduced a new `delegation-check` skill to validate command delegation prompts against agent role definitions, ensuring content separation and conflict detection.
- Established comprehensive separation rules for command delegation prompts and agent definitions, detailing ownership and conflict patterns.
- Improved documentation for command and agent design specifications to reflect the updated spawning patterns and validation processes.

.claude/skills/prompt-generator/specs/agent-design-spec.md

@@ -4,7 +4,7 @@ Guidelines for Claude Code **agent definition files** (role + domain expertise).
 
 ## Content Separation Principle
 
-Agents are spawned by commands via `Task()`. The agent file defines WHO the agent is and WHAT it knows. It does NOT define WHEN or HOW it gets invoked.
+Agents are spawned by commands via `Agent()`. The agent file defines WHO the agent is and WHAT it knows. It does NOT define WHEN or HOW it gets invoked.
 
 | Concern | Belongs in Agent | Belongs in Command |
 |---------|-----------------|-------------------|

.claude/skills/prompt-generator/specs/command-design-spec.md

@@ -153,15 +153,15 @@ Display banners before major phase transitions (agent spawning, user decisions,
 
 ## Agent Spawning Pattern
 
-Commands spawn agents via `Task()` with structured prompts:
+Commands spawn agents via `Agent()` with structured prompts:
 
-```markdown
-Task(
-  prompt=filled_prompt,
-  subagent_type="agent-name",
-  model="{model}",
-  description="Verb Phase {X}"
-)
+```javascript
+Agent({
+  subagent_type: "agent-name",
+  prompt: filled_prompt,
+  description: "Verb Phase {X}",
+  run_in_background: false
+})
 ```
 
 ### Prompt Structure for Agents

.claude/skills/prompt-generator/specs/conversion-spec.md

@@ -49,7 +49,7 @@ Conversion Summary:
 | `## Auto Mode` / `## Auto Mode Defaults` | `<auto_mode>` section |
 | `## Quick Reference` | Preserve as-is within appropriate section |
 | Inline `AskUserQuestion` calls | Preserve verbatim — these belong in commands |
-| `Task()` / agent spawning calls | Preserve verbatim within process steps |
+| `Agent()` / agent spawning calls | Preserve verbatim within process steps |
 | Banner displays (`━━━`) | Preserve verbatim |
 | Code blocks (```bash, ```javascript, etc.) | **Preserve exactly** — never modify code content |
 | Tables | **Preserve exactly** — never reformat table content |