Is it possible to override default
update_count_callback for category? The default function counts published posts while I want to count private posts. I do not want to register a new taxonomy.
In your update_count_callback function do a check for $post->post_status and don't increment your count if post_status is not private.
See this excellent answer on writing a custom update_count_callback callback function.
Misread the question. To override the existing default for the category taxonomy you can create a function that overrides the global $wp_taxonomies variable