saving/reading custom field value does not work – no value gets POSTed

You have two issues:

1.- Your are not passing the correct post ID. To solve this problem change:

update_post_meta( $post->ID, 'my_key', $mydata );

To:

update_post_meta( $post_id, 'my_key', $mydata );

2.- Your are not createing the input correctly, the name attribute is missed:

Change:

echo '<input type="text" value="' . $value . '" id="my_id">';

To:

echo '<input type="text" value="' . $value . '" id="my_id" name="my_id">';