Legacy widgets lack access to some usual server-side globals

[noisysocks]

Copied from ​https://github.com/WordPress/gutenberg/issues/33404.

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## Description

Legacy widgets whose behavior depends on the $pagenow global might not render as expected via the new /wp/v2/widget-types/fm-demo/encode REST API endpoint because $pagenow during these requests is index.php, not widget.php.

Additionally, legacy widgets are no longer able to rely on the $_POST['action'] variable during requests to save widget data, which is usually set to save-widget, and these widgets might not be able to save data as expected.

Similarly, legacy widgets are no longer able to rely on DOING_AJAX or wp_doing_ajax(), although that is typical of the REST API.

## Step-by-step reproduction instructions

The easiest way to observe this behavior is to observe $pagenow and $_POST using Xdebug with a breakpoint inside of WP_REST_Widget_Types_Controller::encode_form_data().

To see the effect of the behavior on a real-world library:

1. Clone and include the Fieldmanager library: ​https://github.com/alleyinteractive/wordpress-fieldmanager
2. Clone and include the Fieldmanager Widgets extension: ​https://github.com/alleyinteractive/fm-widgets/
3. Load the demo widget included in the README: ​https://github.com/alleyinteractive/fm-widgets/blob/296d5aef0585f3ff9bcf223e30867e069c13e2ca/README.md
4. Step through the \fm_widgets_calculated_context() function, which relies on all of the signals mentioned in the description.

## Expected behaviour

With respect to legacy widgets, $pagenow will be widgets.php during form rendering, and $_POST['action'] will be save-widget during saving.

## Actual behaviour

$pagenow is index.php, and $_POST['action'] is unset.

## Code snippet (optional)

See README link above.

## WordPress information

• WordPress version: 5.8-RC2
• Gutenberg version: Not installed
• Are all plugins except Gutenberg deactivated? No, see repro steps.
• Are you using a default theme (e.g. Twenty Twenty-One)? Yes

## Device information

• Device: Desktop
• Operating system: macOS 10.14
• Browser: Chrome 91

[noisysocks]

I'm okay with spoofing these variables for the REST API endpoints that call into WP_Widget::widget and WP_Widget::update.

global $pagenow;
$pagenow = 'widgets.php';
$_POST['action'] = 'save-widget';