Check the config file of the website, maybe the problem is given by these directives:
Options Indexes FollowSymLinks MultiViews
allow from all
Now, if MultiViews is enabled (i.e. without the minus) if you digit /page the server will attemp to find a file page.html or a directory /page/ inside the server root and this is probably the reason why your rewrite rule seems to work properly with html files.
Secondly, usually in fresh installs AllowOverride is set to None, in these cases the server will ignore the .htaccess file, so try to change it as the above, to do that disable the website:
sudo a2dissite website
browse to /etc/apache2/sites-available/ and modify the file, then reanable and reload the server: