My try to simplify and prettify the method.

Avoid using globals in async code, pass Token as argument.

// optimized out async State Machine
private Task UpdateLogsAsync(IProgress<string> status, CancellationToken token) => Task.Run(async () =>
{
    await foreach (var message in this.Server.GetLogMessages().WithCancellation(token))
    {
        token.ThrowIfCancellationRequested();
        status.Report(message);
    }
}, token);

Usage

// synchronized callback. Create new in UI Thread and its body will be always executed there.
IProgress<string> status = new Progess<string>(message =>
{
    this.textLogs.AppendText($"{message}{Environment.NewLine}");
});
try
{
    await UpdateLogsAsync(status, this.CancellationTokenSource.Token);
}
catch (OperationCanceledException ex)
{
    status.Report(ex.Message);
}

My try to simplify and prettify the method.

Avoid using globals in async code, pass Token as argument.

// optimized out async State Machine
private Task UpdateLogsAsync(IProgress<string> status, CancellationToken token) => Task.Run(async () =>
{
    await foreach (var message in this.Server.GetLogMessages().WithCancellation(token))
    {
        status.Report(message);
    }
}, token);

Usage

// synchronized callback. Create new in UI Thread and its body will be always executed there.
IProgress<string> status = new Progess<string>(message =>
{
    this.textLogs.AppendText($"{message}{Environment.NewLine}");
});
try
{
    await UpdateLogsAsync(status, this.CancellationTokenSource.Token);
}
catch (OperationCanceledException ex)
{
    status.Report(ex.Message);
}

My try to simplify and prettify the method.

Avoid using globals in async code, pass Token as argument.

// optimized out async State Machine
private Task UpdateLogsAsync(IProgress<string> status, CancellationToken token) => Task.Run(async () =>
{
    await foreach (var message in this.Server.GetLogMessages())
    {
        token.ThrowIfCancellationRequested();
        status.Report(message);
    }
}, token);

Usage

// synchronized callback. Create new in UI Thread and its body will be always executed there.
IProgress<string> status = new Progess<string>(message =>
{
    this.textLogs.AppendText($"{message}{Environment.NewLine}");
});
try
{
    await UpdateLogsAsync(status, this.CancellationTokenSource.Token);
}
catch (OperationCanceledException ex)
{
    status.Report(ex.Message);
}

My try to simplify and prettify the method.

Avoid using globals in async code, pass Token as argument.

// optimized out async State Machine
private Task UpdateLogsAsync(IProgress<string> status, CancellationToken token) => Task.Run(async () =>
{
    await foreach (var message in this.Server.GetLogMessages().WithCancellation(token))
    {
        token.ThrowIfCancellationRequested();
        status.Report(message);
    }
}, token);

Usage

// synchronized callback. Create new in UI Thread and its body will be always executed there.
IProgress<string> status = new Progess<string>(message =>
{
    this.textLogs.AppendText($"{message}{Environment.NewLine}");
});
try
{
    await UpdateLogsAsync(status, this.CancellationTokenSource.Token);
}
catch (OperationCanceledException ex)
{
    status.Report(ex.Message);
}