Fixes to auto-download/start azsdk mcp (#6578)

Co-authored-by: Ben Broderick Phillips <bebroder@microsoft.com>

eng/common/mcp/azure-sdk-mcp.ps1

@@ -1,7 +1,7 @@
 #!/bin/env pwsh
 
 param(
-    [string]$FileName = 'azsdk',
+    [string]$FileName = 'Azure.Sdk.Tools.Cli',
     [string]$Package = 'azsdk',
     [string]$Version, # Default to latest
     [string]$InstallDirectory = (Join-Path $HOME ".azure-sdk-mcp" "azsdk"),
@@ -56,5 +56,5 @@ $exe = Install-Standalone-Tool `
     -Repository $Repository
 
 if ($Run) {
-    Start-Process -FilePath $exe -NoNewWindow -Wait
-}
+    Start-Process -FilePath $exe -ArgumentList 'start' -NoNewWindow -Wait
+}