Magento 2 – How to Add Action on New Button in Backend Form Container

Add below code in app\code\Vendor\ModuleName\Block\Adminhtml\Module\Edit.php in _construct()

protected function _construct(){
        $this->addButton(
            'delete',
            [
                'label' => __('Delete'),
                'onclick' => 'deleteConfirm(' . json_encode(__('Are you sure you want to do this?'))
                    . ','
                    . json_encode($this->getDeleteUrl()
                    )
                    . ')',
                'class' => 'scalable delete',
                'level' => -1
            ]
        );
}

Below will be your delete action.

/**
 * @param array $args
 * @return string
 */
public function getDeleteUrl(array $args = [])
{
    $params = array_merge($this->getDefaultUrlParams(), $args);
    return $this->getUrl('module/*/delete', $params);
}

I don't have a full explanation, but I have an idea.
All elements rendered with mage-init will/should be handled by some javascript code.
You are linking these buttons to a form, and the form is handled by this js file lib/web/mage/backend/form.js where a jquery ui widget is created.

These are the options for the widget

options: {
    handlersData: {
        save: {},
        saveAndContinueEdit: {
            action: {
                args: {
                    back: 'edit'
                }
            }
        },
        preview: {
            target: '_blank'
        }
    }
},

you can see a saveAndContinueEdit in there somewhere inside handlersData.
Looking for the usages of handlersData you end up in _beforeSubmit where some magic happens (I don't really understand everything in there), and at one point _processData is called.
Moving to _processData you will see something like this

if (attrName === 'action') {
    data[attrName] = this._getActionUrl(attrValue);
}

this means that the action of the form is changed based on the button pressed.

the _getActionUrl function looks like this

_getActionUrl: function(data) {
    if ($.type(data) === 'object') {
        return this._buildURL(this.oldAttributes.action, data.args);
    } else {
        return $.type(data) === 'string' ? data : this.oldAttributes.action;
    }
},

you can see in there data.args involved. The same variable in the widget options for saveAndContinueEdit.

Conclusion: When you set the role saveAndContinueEdit toa submit button, the action of the form is changed via js and back/edit is added to the url.

on_click is transformed to the onclick event and simply called.
I honestly have no idea why there are 2 ways of doing this. Maybe the delete actions didn't get refactored yet.