The longer the commute, the more beneficial cycling specific shoes will be. Cycling shoes are specifically built with very stiff soles to minimize foot flex (and fatigue) and to maximize power transfer.

As to whether a MTB style shoe (recessed cleat), or conventional road shoe (non recessed cleat) is best, it depends on how far you have to walk to your desk or work station once you get off your bike.

Since you state that you're using straps, I'd probably go with MTB shoes as they are easier to walk in while off the bike.

If your commute is less than 5 miles, it probably doesn't matter..... except, as the weather gets nicer, you may find you want to tack on some extra miles on the way home. A cycling shoe will make this more enjoyable. If you live in a hot climate, look for a shoe that is well ventilated.