Unify publishing flow between post and site editor - technical refinements

[ntsekouras]

This issue can be considered as the second part of: #59878 in order to properly fully unify the publishing flow between post and site editors.

Currently the publishing flow seems and is similar in terms of UI, but we still need to have some follow ups to consolidate some functionality in a better way.

The extra use cases in site editor include:

1. Fix duplicate saving panels. This is related to the next item and we need to look into how to best consolidate save panels and button in editor package.
2. Activating/previewing a theme (change of labels, callbacks, etc..). See customSaveButton in Editor component.
3. The save button in site editor's sidebar (view mode) renders the save panel in a Modal
4. Some specific logic for how to handle the save when making changes to global styles page
5. Some cleanup is needed and let's refactor a bit where we can, while doing these follow ups (ex)

Let's track the refinement of the flow and UI in a separate issue to keep the technical improvement separate.