How to show only the posts that belong to a specific category?

Adding on to the code provided in your previous post, this should do the trick:

<?php
$args = array(
    'type'        => 'childrenscakes',
    'taxonomy'    => 'childrenscakes-category',
);

$categories = get_categories( $args );
foreach($categories as $cat) {
    echo '<a href="' . get_category_link($cat->term_id) . '">' . $cat->name . '</a><br/>';
}
?>

Please review get_categories() from the Codex for more examples.

Could I use custom taxonomies (instead of categories)?

get_term_link( $term, $taxonomy = '')