The 5k Training app works slightly differently to the standard Zombies, Run! app so we refer to missions in this app as 'Workouts'. These are separate from missions in Zombies, Run! so you see them on ZombieLink on a different page. In order to see the runs you've logged in the 5k app you'll need to go to the 5k Training area of your account.
Finally, 'The Story Begins...' is included in both apps, so you will need to re-do it in the main Zombies, Run! app in order for this to be marked as completed on the Missions page.