Why doesn't custom mysql query return results? Is syntax correct?

First time making a custom query. I was looking for a way to get a list of categories for a certain taxonomy so I could run a LIKE query for a autocomplete function. Just to get a few results I came up with the mysql query that gives me results in phpMyAdmin

SELECT * 
FROM wp_terms
INNER JOIN wp_term_taxonomy ON ( wp_terms.term_id = wp_term_taxonomy.term_id ) 
WHERE wp_terms.name LIKE  'ca%'
AND count >0
LIMIT 0 , 30

My wordpress function looks like this

function get_categories_by_like_searching() {
    global $wpdb;
    $keyword = $_GET['query']; // example : "ca" would return categories with name of "cats"
    $query = $wpdb->prepare( "SELECT * FROM $wpdb->wp_terms 
                                INNER JOIN $wpdb->wp_term_taxonomy 
                                    ON ($wpdb->wp_terms.term_id = $wpdb->wp_term_taxonomy.term_id) 
                                WHERE $wpdb->wp_terms.name LIKE %s AND $wpdb->wp_terms.count > 0", $wpdb->esc_like($keyword) . '%');
    $categories = $wpdb->query($query);
    foreach($categories as $category) { 
        array_push($results,$category['name']);
    }
    $results = array("suggestions"=>$results);
    $response = json_encode($results);
    // response output
    header( "Content-Type: application/json" );
    echo $response;
    exit;
}

Am I making the connection incorrectly or is my wordpress query syntax off?

The exact function I'm looking for is really this:

<?php 

$args = array(
    'taxonomy'                 => 'my_custom_taxonomy'
     // with category names LIKE 'x%'

); 
$categories = get_categories($args);
?>