Agent Teams: allow team lead to specify process display names for teammates

Feature Request

When using Agent Teams, the team lead can assign meaningful names to teammates via the Task tool's name parameter (e.g., phase-1-c3-PM, phase-1-c3-coding). These names work correctly for SendMessage routing within the team.

However, the process display names shown in the Claude Desktop/Web sidebar are auto-generated by the platform using a {hostname}-{random-adjective}-{random-noun} pattern (e.g., vm-dev-whale-01-steady-hinton). These names are not recognizable and make it difficult to identify which teammate is which when monitoring a build.

Current Behavior

• Team lead spawns teammates with specific names
• SendMessage routing uses the correct names
• Sidebar display shows auto-generated random names unrelated to the teammate names

Desired Behavior

The process display name in the sidebar should reflect the name the team lead assigned when spawning the teammate, or the team lead should have a way to specify the display name explicitly.

Use Case

In structured workflows (e.g., a PM agent and a Coding agent working on a component build), the sponsor monitors the build by watching the sidebar. Random names like vm-dev-whale-01-staged-sparkle provide no signal about which agent is doing what. Recognizable names like phase-1-c3-PM and phase-1-c3-coding would make monitoring practical.

Environment

• Claude Code with CLAUDE_CODE_EXPERIMENTAL_AGENT_TEAMS=1
• Running on a Linux dev VM via SSH
• Teammates spawned via Task tool with team_name and name parameters