Fix: Update error message in private actions.

[kartikmehta8]

What?

Why?

This PR improves error handling in the Gutenberg editor by providing more specific and user-friendly error messages when saving operations fail. It replaces the generic "Saving failed" message with detailed context, making it easier to identify the issue and troubleshoot.
Closes #57245

How?

The current "Saving failed" message is too generic and does not provide sufficient information for users or developers to understand the root cause of the error.

Testing Instructions

1. Open the browser's Developer Tools and block some API requests in the Network tab.
2. Save changes and verify that the message "Some records could not be saved" is displayed.

[github-actions]

The following accounts have interacted with this PR and/or linked issues. I will continue to update these lists as activity occurs. You can also manually ask me to refresh this list by adding the props-bot label.

If you're merging code through a pull request on GitHub, copy and paste the following into the bottom of the merge commit message.

Co-authored-by: kartikmehta8 <kartikmehta@git.wordpress.org>
Co-authored-by: Mamaduka <mamaduka@git.wordpress.org>
Co-authored-by: creativecoder <grantmkin@git.wordpress.org>

To understand the WordPress project's expectations around crediting contributors, please review the Contributor Attribution page in the Core Handbook.

[github-actions]

👋 Thanks for your first Pull Request and for helping build the future of Gutenberg and WordPress, @kartikmehta8! In case you missed it, we'd love to have you join us in our Slack community.

If you want to learn more about WordPress development in general, check out the Core Handbook full of helpful information.

[Mamaduka]

Thanks for contributing, @kartikmehta8!

Somehow, I missed that #57245 is similar to #64158 when closing duplicate issues. I created a proposal last month, but the discussion is still ongoing.