I finally got an answer to this problem from my hosting provider …
It seems that the installer script has been adding a line into .htaccess, which tells the server to use PHP 5.2 in that directory and all subfolders:
AddHandler application/x-httpd-php52 .php .php5 .php4 .php3
You can simply remove it and save over, in the .htaccess file for the site’s directory–or comment it out, by adding a # so it looks like:
#AddHandler application/x-httpd-php52 .php .php5 .php4 .php3