Add the rest of the changes that were supposed to be in the last commit

- Get most of the changes that were supposed to be in the last commit
- Go back to not processing messages until IMcpServer.RunAsync is called

Failed test run: https://github.com/modelcontextprotocol/csharp-sdk/actions/runs/14299175970/job/40070612363?pr=225

Author: halter73

src/ModelContextProtocol/Client/McpClient.cs

@@ -106,7 +106,10 @@ public async Task ConnectAsync(CancellationToken cancellationToken = default)
         {
             // Connect transport
             _sessionTransport = await _clientTransport.ConnectAsync(cancellationToken).ConfigureAwait(false);
-            StartSession(_sessionTransport);
+            InitializeSession(_sessionTransport);
+            // We don't want the ConnectAsync token to cancel the session after we've successfully connected.
+            // The base class handles cleaning up the session in DisposeAsync without our help.
+            StartSession(_sessionTransport, fullSessionCancellationToken: CancellationToken.None);
 
             // Perform initialization sequence
             using var initializationCts = CancellationTokenSource.CreateLinkedTokenSource(cancellationToken);

src/ModelContextProtocol/Server/McpServer.cs

@@ -18,6 +18,8 @@ internal sealed class McpServer : McpEndpoint, IMcpServer
         Version = DefaultAssemblyName.Version?.ToString() ?? "1.0.0",
     };
 
+    private readonly ITransport _sessionTransport;
+
     private readonly EventHandler? _toolsChangedDelegate;
     private readonly EventHandler? _promptsChangedDelegate;
 
@@ -41,6 +43,7 @@ public McpServer(ITransport transport, McpServerOptions options, ILoggerFactory?
 
         options ??= new();
 
+        _sessionTransport = transport;
         ServerOptions = options;
         Services = serviceProvider;
         _endpointName = $"Server ({options.ServerInfo?.Name ?? DefaultImplementation.Name} {options.ServerInfo?.Version ?? DefaultImplementation.Version})";
@@ -81,8 +84,8 @@ public McpServer(ITransport transport, McpServerOptions options, ILoggerFactory?
             prompts.Changed += _promptsChangedDelegate;
         }
 
-        // And start the session.
-        StartSession(transport);
+        // And initialize the session.
+        InitializeSession(transport);
     }
 
     public ServerCapabilities? ServerCapabilities { get; set; }
@@ -112,9 +115,8 @@ public async Task RunAsync(CancellationToken cancellationToken = default)
 
         try
         {
-            using var _ = cancellationToken.Register(static s => ((McpServer)s!).CancelSession(), this);
-            // The McpServer ctor always calls StartSession, so MessageProcessingTask is always set.
-            await MessageProcessingTask!.ConfigureAwait(false);
+            StartSession(_sessionTransport, fullSessionCancellationToken: cancellationToken);
+            await MessageProcessingTask.ConfigureAwait(false);
         }
         finally
         {

src/ModelContextProtocol/Shared/McpEndpoint.cs

@@ -5,6 +5,7 @@
 using ModelContextProtocol.Protocol.Transport;
 using ModelContextProtocol.Server;
 using ModelContextProtocol.Utils;
+using System.Diagnostics;
 using System.Diagnostics.CodeAnalysis;
 using System.Reflection;
 
@@ -62,12 +63,16 @@ public IAsyncDisposable RegisterNotificationHandler(string method, Func<JsonRpcN
     /// </summary>
     protected Task? MessageProcessingTask { get; private set; }
 
-    [MemberNotNull(nameof(MessageProcessingTask))]
-    protected void StartSession(ITransport sessionTransport)
+    protected void InitializeSession(ITransport sessionTransport)
     {
-        _sessionCts = new CancellationTokenSource();
         _session = new McpSession(this is IMcpServer, sessionTransport, EndpointName, RequestHandlers, NotificationHandlers, _logger);
-        MessageProcessingTask = _session.ProcessMessagesAsync(_sessionCts.Token);
+    }
+
+    [MemberNotNull(nameof(MessageProcessingTask))]
+    protected void StartSession(ITransport sessionTransport, CancellationToken fullSessionCancellationToken)
+    {
+        _sessionCts = CancellationTokenSource.CreateLinkedTokenSource(fullSessionCancellationToken);
+        MessageProcessingTask = GetSessionOrThrow().ProcessMessagesAsync(_sessionCts.Token);
     }
 
     protected void CancelSession() => _sessionCts?.Cancel();
@@ -122,5 +127,5 @@ public virtual async ValueTask DisposeUnsynchronizedAsync()
     }
 
     protected McpSession GetSessionOrThrow()
-        => _session ?? throw new InvalidOperationException($"This should be unreachable from public API! Call {nameof(StartSession)} before sending messages.");
+        => _session ?? throw new InvalidOperationException($"This should be unreachable from public API! Call {nameof(InitializeSession)} before sending messages.");
 }