Change the user is logged in cookie name

I have menu items that are set to be shown/hidden depending on if the user is logged in. I would like to change the default cookie name that WP looks for to determine login status.

My cookie name is _example and has a value of 1 when user is logged in.

I have tried define( 'TEST_COOKIE', '_example' ); in wp-config.php, but that didn’t seem to do the trick.

How can I replace the default cookie name?

Note: I do not need to identify which user is logged in, only whether or not it is true to show/hide menu items. _example has a value of 1 if logged in, otherwise _example is not present or 0.