I'd go for public_html first, "most likely suspect."
Servers are generally configured to look for indexing in a particular order, like index.html, if not found, index.html, if not found . . . others, depends on what's in your configuration. So index.html should be fine - but a small hint, for links back to "home page," don't do
This gives you the freedom to do things with "home" that may or may not involve changing the extension, without having to consider SEO issues.