Problem

Background processes are stored in a Map<processId, BackgroundProcess> in BackgroundProcessManager. The processId was set directly from displayName, and it's also used to compute the output directory:

const processId = config.displayName;
// ...
const outputDir = `${bgOutputDir}/${workspaceId}/${processId}`;

When a new process is spawned in the background with a display name that was previously used:

1. It gets the same outputDir as the old process
2. The old exit_code file still exists in that directory (written by shell trap on exit)
3. When refreshRunningStatuses() checks the new process, it reads the stale exit_code file
4. The new process is immediately marked as "exited" even though it just started
5. The banner filters to status === "running" → process doesn't appear

Additionally, the map entry itself gets overwritten, losing tracking of any previous process with that name.

Fix

Append incrementing suffix (1), (2), etc. when a collision is detected:

let processId = config.displayName;
let suffix = 1;
while (this.processes.has(processId)) {
  processId = `${config.displayName} (${suffix})`;
  suffix++;
}

This ensures each process gets a unique outputDir, so no stale exit_code files interfere.

The original displayName is preserved separately on the BackgroundProcess object for UI display - the banner shows clean names like "Dev Server" while the agent receives unique IDs like "Dev Server (1)" to reference specific processes.

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