You can use the post type within the action name and you gain the removing of an unnecessary check of the post type.
More on this wp_transition_post_status
For instance: When publishing a post for the first time, the post
status may transition from ‘draft’ – or some other status – to
‘publish’. However, if a post is already published and is simply being
updated, the “old” and “new” statuses may both be ‘publish’ before and
after the transition.
add_action( 'publish_flamingo_inbound', 'my_update_flamingo_inbound_author', 10);
And removing the check for post type:
if($post_type == 'flamingo_inbound') {