update flamingo_inbound post type after insert

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') {