After copying over the files for 7.31 (was 7.26), minus the /sites folder, and running update.php, the result is "no pending updates." I followed the instructions at https://www.drupal.org/node/1494290 properly - any thoughts about what to do next? Thanks.
Go to '[your-website]/admin/reports/status' and if there you see that Drupal version is 7.31 (which you probably will see), if there is no some other errors and everything works fine on website you successfully did update.
It is not necessary to always have database update if you are updating some module or Drupal itself.