Magento – How to discover the XML Layout name for a page

Since you are using your own controller you don't need to only use the default handles. Based on the logged in status you could add in your infoAction method

 $this->getLayout()->getUpdate()->addHandle('mymodule_customer_info_logged_in');

or

 $this->getLayout()->getUpdate()->addHandle('mymodule_customer_info_logged_out');

and then in your layout.xml file use

<mymodule_customer_info_logged_in>

and

<mymodule_customer_info_logged_out>

-- addition after your edit below --

Seeing that you don't want to edit the controller (since it is a 3rd party extension), I would create a separate extension that only observes controller_action_layout_load_before

    $update = $observer->getEvent()->getLayout()->getUpdate();
    $handles = $update->getHandles();

    if (in_array('custommodule_customer_info', $handles)) {

        //code to add the custom handles based on login
    }

You can't do that. Specifically, you can't use the <update/> tag in a CMS page's Layout Update XML field.

Magento processes the <update/> nodes when it is initially loading the layout update files, and layout updates from the core_layout_update table. This is done with the fetchRecursiveUpdates function.

#File: app/code/core/Mage/Core/Model/Layout/Update.php

public function fetchPackageLayoutUpdates($handle)
{
    //...
        $this->fetchRecursiveUpdates($updateXml);
    //...

    return true;
}

public function fetchDbLayoutUpdates($handle)
{
    //...
    $this->fetchRecursiveUpdates($updateXml);        
    //...        
}

//...
public function fetchRecursiveUpdates($updateXml)
{
    foreach ($updateXml->children() as $child) {
        if (strtolower($child->getName())=='update' && isset($child['handle'])) {
            $this->merge((string)$child['handle']);
            // Adding merged layout handle to the list of applied hanles
            $this->addHandle((string)$child['handle']);
        }
    }
    return $this;
}

A CMS page, however, is rendered later on via a helper method. The rendering of a CMS page includes this code

#File: app/code/core/Mage/Cms/Helper/Page.php

$action->loadLayoutUpdates();
$layoutUpdate = ($page->getCustomLayoutUpdateXml() && $inRange)
    ? $page->getCustomLayoutUpdateXml() : $page->getLayoutUpdateXml();
$action->getLayout()->getUpdate()->addUpdate($layoutUpdate);
$action->generateLayoutXml()->generateLayoutBlocks();    

The loadLayoutUpdates call is the standard loading of the layout, including the fetchRecursiveUpdates mentioned above.

Then, Magento looks for a custom layout XML update set on the CMS page object

$layoutUpdate = ($page->getCustomLayoutUpdateXml() && $inRange)
    ? $page->getCustomLayoutUpdateXml() : $page->getLayoutUpdateXml();

and adds it directly to the list of updates to apply

$action->getLayout()->getUpdate()->addUpdate($layoutUpdate);

This means the scanning for the update tag is never done.

If I was trying to implement something like this, I'd add a new block in the CMS page's layout update XML, and then add the shared layout items programmatically in the blocks _prepareLayout method. (not sure off the top of my head if that would work w/r/t to timing, but that'd be where I'd start)