Showing custom taxonomy column in taxonomy category listings immediately after create category

What is happening is that:

  • the page /wp-admin/edit-tags.php uses $_GET['taxonomy']

  • and /wp-admin/admin-ajax.php, fired when we create a new term, uses $_POST['taxonomy']

To solve it, use $_REQUEST in manage_my_category_columns().
It will cover both GET and POST:

if ( !isset( $_REQUEST['taxonomy']) || $_REQUEST['taxonomy'] != 'products' )

Related Q&A:
WP List Table custom quick edit box – post meta data missing and columns change on submit