The template naming is a bit confusing in this situation, but what is happening is normal behavior in the Template Hierarchy.
home.php shows your posts page, whether it’s on the front page, or assigned to another page when using a static front page. If you want your home.php template to be used for your static front page, rename it front-page.php. In the absence of a home.php template, your posts page (Press) will use the index.php template.