CmdPal: Add filter by Terminal channel (#41582)

## Summary of the Pull Request

- Introduces a new filter on the Profiles page to filter by Terminal
channel.
- Adds a new option to remember the last select Terminal channel (or
automatically reset to All Channels).
- Adds new classes `AppSettings` and `AppSettingsManager` to hold
non-user settings.

Pictures? Pictures!

<img width="1485" height="931" alt="image"
src="https://github.com/user-attachments/assets/2cec7a8d-efe6-4692-a7ba-9608fb181624"
/>

<img width="1730" height="1014" alt="image"
src="https://github.com/user-attachments/assets/87984b82-e085-42a5-b71c-5ddc71ff52ec"
/>

<img width="1722" height="1063" alt="image"
src="https://github.com/user-attachments/assets/97baff23-3db0-404b-8a8d-622f841b344b"
/>

<!-- Please review the items on the PR checklist before submitting-->
## PR Checklist

- [x] Closes: #41432 
- [ ] **Communication:** I've discussed this with core contributors
already. If the work hasn't been agreed, this work might be rejected
- [ ] **Tests:** Added/updated and all pass
- [ ] **Localization:** All end-user-facing strings can be localized
- [ ] **Dev docs:** Added/updated
- [ ] **New binaries:** Added on the required places
- [ ] [JSON for
signing](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ESRPSigning_core.json)
for new binaries
- [ ] [WXS for
installer](https://github.com/microsoft/PowerToys/blob/main/installer/PowerToysSetup/Product.wxs)
for new binaries and localization folder
- [ ] [YML for CI
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ci/templates/build-powertoys-steps.yml)
for new test projects
- [ ] [YML for signed
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/release.yml)
- [ ] **Documentation updated:** If checked, please file a pull request
on [our docs
repo](https://github.com/MicrosoftDocs/windows-uwp/tree/docs/hub/powertoys)
and link it here: #xxx

<!-- Provide a more detailed description of the PR, other things fixed,
or any additional comments/features here -->
## Detailed Description of the Pull Request / Additional comments

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wherever possible, but list manual validation steps taken as well -->
## Validation Steps Performed

src/modules/cmdpal/ext/Microsoft.CmdPal.Ext.WindowsTerminal/Helpers/AppSettingsManager.cs

@@ -0,0 +1,72 @@
+// Copyright (c) Microsoft Corporation
+// The Microsoft Corporation licenses this file to you under the MIT license.
+// See the LICENSE file in the project root for more information.
+
+#nullable enable
+
+using System;
+using System.IO;
+using System.Text.Json;
+using ManagedCommon;
+using Microsoft.CommandPalette.Extensions.Toolkit;
+
+namespace Microsoft.CmdPal.Ext.WindowsTerminal.Helpers;
+
+#nullable enable
+
+public sealed class AppSettingsManager
+{
+    private const string FileName = "appsettings.json";
+
+    private static string SettingsPath()
+    {
+        var directory = Utilities.BaseSettingsPath("Microsoft.CmdPal");
+        Directory.CreateDirectory(directory);
+        return Path.Combine(directory, FileName);
+    }
+
+    private readonly string _filePath;
+
+    public AppSettings Current { get; private set; } = new();
+
+    public AppSettingsManager()
+    {
+        _filePath = SettingsPath();
+        Load();
+    }
+
+    public void Load()
+    {
+        try
+        {
+            if (File.Exists(_filePath))
+            {
+                var json = File.ReadAllText(_filePath);
+                var loaded = JsonSerializer.Deserialize(json, AppSettingsJsonContext.Default.AppSettings);
+                if (loaded is not null)
+                {
+                    Current = loaded;
+                }
+            }
+        }
+        catch (Exception ex)
+        {
+            ExtensionHost.LogMessage(new LogMessage { Message = ex.ToString() });
+            Logger.LogError("Failed to load app settings", ex);
+        }
+    }
+
+    public void Save()
+    {
+        try
+        {
+            var json = JsonSerializer.Serialize(Current, AppSettingsJsonContext.Default.AppSettings);
+            File.WriteAllText(_filePath, json);
+        }
+        catch (Exception ex)
+        {
+            ExtensionHost.LogMessage(new LogMessage { Message = ex.ToString() });
+            Logger.LogError("Failed to save app settings", ex);
+        }
+    }
+}