Site Editor: global Save button shows up when not needed

[afercia]

Description

It appears the 'global' Save button added to the Site editor sidebar in #47142 shows up unexpectedly in some flows where it's not actually needed.

In the 'browse mode' templates list, users can take some actions. For example they can:

• Rename a custom template.
• Clear customizations made to a template.

When taking these actions, the 'Save' button appears in the 'saving' state at the bottom of the sidebar. Screenshot:

For better clarity, here's an animated GIF:

The global Save button makes perfectly sense for the scenarios illustrated in #47142 and #46985

I'm not sure it should show up at all when saving changes from the templates list. That's totally unexpected and confusing for users. I'm guessing this potentially happens for all the flows where there's a similar saving process.

Cc @youknowriad

Step-by-step reproduction instructions

• Go to the Site Editor.
• Create a custom template.
• Switch back to 'browse' more and click Templates > Manage all templates.
• From the templates list, click the 'Actions' button related to the custom template you created.
• Choose 'Rename', set a new name for the template and save.
• Observe that while saving, the global 'Save' button unexpectedly shows up in the Sidebar and the button is in the 'saving' state.

Screenshots, screen recording, code snippet

No response

Environment info

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Please confirm that you have searched existing issues in the repo.

Yes

Please confirm that you have tested with all plugins deactivated except Gutenberg.

Yes