Solved it using the second method (check if parent is Home, by checking if page has a Parent count of more than 0).
After reading this:
<?php if ($page->parents()->count() > 0) : ?>
I decided to check using that method (if page has parent count greater than 0). This works, but I’d still love a better method. I like that it works, but I fear it’s using incorrect logic and would break if the site structure would grow beyond this.
Let me know what you think I should use instead.