It turned out, that the application was deployed in development, not production despite the capistrano command and the settings in deploy/production.rb. And in database.yml I had incorrect settings in the development block - they were supposed to work only on my laptop but not at the VPS.
It was a great relief to find the explanation of that mysterious error. But how could be that "cap production deploy" produced a deployment in a development environment, rather than in production? I don't have 100% certainty as I changed a number of things while trying to find the culprit but the most likely reason was the following line in /etc/nginx/nginx.conf:
Once I changed that setting and re-started nginx the problem disappeared. I am relieved but I still find the whole thing a bit troubling. Somehow I cannot escape the feeling that the rails stack is really messy and fragile - you can specify the same setting (in this case the environment) in a number of different places (in fact in different layers of the stack) so that you get inconsistency and unexpected behaviour. God knows what other settings are now over-ridden but haven't yet manifested themselves in an error. This is a bit frightening.
Anyway - you have to live with it if you want to use rails. For those who are suspecting a similar problem in their deployment:
The best way to tell which environment your application is running in is to look at the logs (/home/deploy_user/my_app/shared/log). In my case the file development.log had a current timestamp while production.log was stale (a few weeks old). That sort of nailed it. However, the situation could be a bit more complicated if you are running different instances of the application at different ports - they could run in different environments. Than you need to look up the logs and find which one has the error message that you are getting via the browser.