EDIT

I don’t really know of a solution within WordPress to sort the way you need. I think the best is to make use of php sorting to sort your returned array as needed

Here is how:

1. Get the array of category info as you normally do

   $categories = get_categories($args);
   

$categories will hold an array of the five categories returned from the database

2. You now need to create an array with your desired sort order

   $keys = [1,3,4,2,5];
   

3. You are now going to combine $keys and $categories with array_combine. We will use the values of Skeys for the keys for our new combined array

   $categories_rearranged = array_combine( $keys, $categories );
   

4. At this stage, your array will not be naturally sorted, so with the use of ksort, sort your array so that it is ordered naturally

   ksort($categories_rearranged);
   

5. $categories_rearranged will now be the array that you will make use of and will be passed to your foreach loop

   foreach ( $categories_rearranged as $category ) {
      echo $category->name;
   }
   

ORIGINAL ANSWER

The codex is there to help you. Look at get_categories() and in particular to the following parameters

orderby

(string) Sort categories alphabetically or by unique category ID. The default is sort by name. Valid values:

• id

• name – default

• slug

• count

• term_group

You can see you are passing an invalid parameter to orderby

The include and exclude parameters cannot be used together as well