Now when I click on the "Open cPanel" button I get this info:
A website is a presence on the web − a personal blog, a forum for your community, information about your brick and mortar business, or even a home for your development project − it's yours to do whatever you'd like with. Create a website today using our easy-to-use software installer, our SiteBuilder, or by manually editing or uploading your website files.
You don't have any websites added. Create one and begin your online presence today!
Ther's really no need if you don't want to use the full view cPanel them (cPanelx3 gives access to more advanced tools), you can reach the filemanager from the basic theme as well, even if the site is deleted from the front (which as you have experience does not delete the actual files for the site, just how it is displayed in the control panel).
Under domains, you'll see what domains you have. At the very least you should see your main domain listed, with a link that takes you to the public_html folder by using the filemanager.
Deleting public_html will delete all public website content (unless you have sub-domains set to folders outside of public_html but your probably don't, not sure if cPanel even allows this and out of sub-domains to test). You can always create a new public_html folder if need be.