The same as above but replacing 'projects' for array('post') should work.

function wpse_287931_register_categories_names_field()
{
    register_rest_field(
        array('post'),
        'categories_names',
        array(
            'get_callback'    => 'wpse_287931_get_categories_names',
            'update_callback' => null,
            'schema'          => null,
        )
    );
}

add_action('rest_api_init', 'wpse_287931_register_categories_names_field');

function wpse_287931_get_categories_names($object, $field_name, $request)
{
    $formatted_categories = array();

    $categories = get_the_category($object['id']);

    foreach ($categories as $category) {
        $formatted_categories[] = $category->name;
    }

    return $formatted_categories;
}

The same as above but replacing 'project' for array('post') should work.

function wpse_287931_register_categories_names_field()
{
    register_rest_field(
        array('post'),
        'categories_names',
        array(
            'get_callback'    => 'wpse_287931_get_categories_names',
            'update_callback' => null,
            'schema'          => null,
        )
    );
}

add_action('rest_api_init', 'wpse_287931_register_categories_names_field');

function wpse_287931_get_categories_names($object, $field_name, $request)
{
    $formatted_categories = array();

    $categories = get_the_category($object['id']);

    foreach ($categories as $category) {
        $formatted_categories[] = $category->name;
    }

    return $formatted_categories;
}