Running drush entity:updates claimed successful but not changing anything
For deploying entity schema updates use an update hook and the entity definition update manager. See the release note of Drupal 8.7.0 and the change record Support for automatic entity updates has been removed.

drush entup is moving from core to contrib: https://www.drupal.org/project/devel_entity_updates


  This project aims to restore this functionality as a developer-only
  tool. This module depends on Devel and is not meant to be enabled in
  production environments or relied upon in deployment workflows.




"Or is there any alternative way to update entity definition and make changes to database?"

Yes, there is. Don't use base fields other than for the very basic things that never change (if core makes changes necessary you'll probably find then examples how to write the update hook). Instead use configurable fields (bundle fields), so that you can deploy easily through configuration.