Notifications: Refresh admin menu after changing comment status

[phcp]

Closes #94909

Proposed Changes

• Refresh admin menu after updating comment status
  • It follows the same approach as the actions on the Comments page (client/my-sites/comments/comment/comment-actions.jsx:87)
  • It needed to include action handlers on the Calypso client, as the Notifications component is implemented on the apps directory and doesn't share the same store

Why are these changes being made?

• Updating comment status doesn't update the comments count on the admin menu; it includes approving, marking as spam, and deleting

Testing Instructions

1. Apply this PR to your local or use a test env
2. Update site setting to hold comments for moderation (Settings > Discussion > A comment is held for moderation)
3. Go to the Calypso home for the testing site (or any other page inside Calypso)
4. On an incognito tab (or another session), comment on a post from the test site
5. Open the notifications panel
6. You should see a notification for the new comment
7. Click on the notification and approve the comment
8. The comment count on the admin menu (sidebar) should update
9. Comment again and now click on Spam in the notification
10. The count on the admin menu (sidebar) should update after the undo state completes (~5 seconds later)
11. Do the same for the Trash action, and the count should also update as expected

Pre-merge Checklist

• Has the general commit checklist been followed? (PCYsg-hS-p2)
• Have you written new tests for your changes?
• Have you tested the feature in Simple (P9HQHe-k8-p2), Atomic (P9HQHe-jW-p2), and self-hosted Jetpack sites (PCYsg-g6b-p2)?
• Have you checked for TypeScript, React or other console errors?
• Have you used memoizing on expensive computations? More info in Memoizing with create-selector and Using memoizing selectors and Our Approach to Data
• Have we added the "[Status] String Freeze" label as soon as any new strings were ready for translation (p4TIVU-5Jq-p2)?
• For changes affecting Jetpack: Have we added the "[Status] Needs Privacy Updates" label if this pull request changes what data or activity we track or use (p4TIVU-aUh-p2)?

[github-actions]

Calypso Live (direct link)

https://calypso.live?image=registry.a8c.com/calypso/app:build-122386

Jetpack Cloud live (direct link)

https://calypso.live?image=registry.a8c.com/calypso/app:build-122386&env=jetpack

Automattic for Agencies live (direct link)

https://calypso.live?image=registry.a8c.com/calypso/app:build-122386&env=a8c-for-agencies

[matticbot]

Here is how your PR affects size of JS and CSS bundles shipped to the user's browser:

Async-loaded Components (~123 bytes added 📈 [gzipped])

name                              parsed_size           gzip_size
async-load-calypso-notifications       +408 B  (+0.2%)     +123 B  (+0.2%)

React components that are loaded lazily, when a certain part of UI is displayed for the first time.

Legend

What is parsed and gzip size?

Parsed Size: Uncompressed size of the JS and CSS files. This much code needs to be parsed and stored in memory.
Gzip Size: Compressed size of the JS and CSS files. This much data needs to be downloaded over network.

Generated by performance advisor bot at iscalypsofastyet.com.

[matticbot]

This PR modifies the release build for the following Calypso Apps:

For info about this notification, see here: PCYsg-OT6-p2

• notifications

To test WordPress.com changes, run install-plugin.sh $pluginSlug fix/comments-stale-count on your sandbox.

[andregardi]

Looks good to me.
We had a long slack discussion on that p1729519003872409-slack-C02TCEHP3HA, but seems to be resolved now.