By default, only users with the `edit_theme_options` capability can access the sidebars and widgets REST API endpoints. In this commit, A new `show_in_rest` parameter is added to the `register_sidebar` function. When enabled, all users will be able to access that sidebar and any widgets belonging to that sidebar.
This commit reduces the `context` for a widget's `instance` information to only `edit`. This is to ensure that internal widget data is not inadvertently exposed to the public. A future ticket may expose additional APIs to allow widget authors to indicate that their instance data can be safely exposed. REST API consumers intending to access this `instance` information should take care to explicitly set the `context` parameter to `edit`.
Props spacedmonkey, zieladam.
Fixes#53915.
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