Awesome! It works!

Thanks for clarifying that. I used Webdoc's code but put it in mysite/code/homepage.php

The filename of mysite/code/homepage.php doesn't have to match the layout filename themes/mytheme/templates/layout/homepage.ss right?

No, the filename can be anything. The file itself can be almost anywhere. The project dir can be renamed (from mysite) through the _config.php $project variable.
Templates can be stored in either a theme (themes/<themename>/templates/) or the module (your project is technically a module too! - <mysite>/temlpates)

Just the class name and the template filename must match.
eg. if for instance you had themes/yourtheme/templates/HomePage.ss, then this would be used as the 'main' template, the 'Layout' would then be HomePage or Page, but only if there is a $Layout var present in the 'main'.

In this way templates sort of mirror the class heirarchy in the manner they're searched for/used.

It can be all a bit confusing when you're just getting used to it, but it makes a lot of sense once you do :)