I'm sure that it's easy once you know how to do it, but how do I change site.com/zenphoto to site.com/gallery?

I've tried simply renaming the zenphoto folder and changing
define('WEBPATH', '/gallery');
define('SERVERPATH', '/usr/local/www/vardefitnessdk/www/gallery');
in zp-config.php accordingly, but though the main page that contains the albums' thumbnails work, the individual album does not work.

Can anybody please help? Thanks in advance!

The way to do is to rename the folder and then run site.com/gallery/zp-core/setup.php again. THis will also change the rewrite base of the .htaccess file (in case you use mod rewrite).

Thank you very much, acrylian. I should have known that it was easy as this. It now works perfectly!

Will the same procedure work if I shift my gallery from url.com/gallery to gallery.url.com ?

Yes, create the subdomain, link it to the /gallery folder and rerun setup.