Many advice me to do the exercises for shoulders, lats, chests first and then do the arm specific ones (biceps, triceps) or the abdominal ones. For eg the full-body workout from the exrx site. Is there any reason why is it necessary to maintain this order?
You use your biceps and triceps to do the other exercises, even though they're not the primary muscles, so if you've already fatigued them, then you won't be able to lift as much weight and won't adequately stimulate the shoulder muscles.
The first selection shows the workouts loosely in the order of largest muscles to smallest, the second looks quite arbitrary to me.
Working larger muscles first is also important as working them stimulates the most testosterone production for later working the smaller muscles. So you should really be working the deadlift or squat first, before working any other muscle groups.