Something must have gone wrong. I did not just upload the WordPress installation, as I have installed WordPress on the domain for about 3 years now.
The last thing I did yesterday, before I saw this install.php page was to click on the UPDATE link for a contact form plugin (Fast Secure Contact Form) in my admin dashboard and what I saw thereafter was that page – that was why I was surprised.
Does this mean my initial WP installation has been erased? Someone please help.