The advice in the manual is correct. The other advice is in error. The reason it is important to apply the parking brake BEFORE shifting the selector into Park, is that once the parking brake is applied, the car will not be able to roll. If the gear selector is put into Park WITHOUT the parking brake being set, as soon as you take your foot off the service brake, the car will still be able to roll a small distance, and this can put pressure on the parking pawl in the transmission.
The parking pawl is a device which locks the transmission gears so that they cannot turn; but there is some free play in that mechanism. It does not stop the car immediately, like the parking brake does. If the car rolls after it is shifted into Park, then it will move until it presses against the locked transmission gears. When the weight of the car presses on the transmission gears; the selector lever, which is connected to those gears, becomes difficult to move. Since the selector lever also connects to the ignition switch lock, the ignition key, in turn, also becomes difficult to move.
If you want to experience an unforgettable demonstration of this, park your car heading up a steep uphill incline, shift the selector into Park; take your foot off the service brake, let the car roll back until it is stopped by the transmission, and then apply the parking brake.
After that, try to start the car. I predict that you'll have so much trouble moving the key, that you'll vow to never do that again. THE ONE DRAWBACK OF THIS DEMONSTRATION IS THE LIKELIHOOD THAT THE PRESSURE OF THE VEHICLE'S WEIGHT, COMPOUNDED BY THE SLOPE OF THE HILL, WILL MAKE IT SO DIFFICULT TO TURN THE KEY THAT YOU MAY HAVE TO CALL A TOW TRUCK TO GET YOU OUT OF THAT SITUATION.
The confusion about which technique is correct largely arises from the experience of drivers who instinctively hold one foot on the service brake until after they have set the parking brake. When it is done that way, the car cannot roll; so it doesn't make any difference whether the selector is shifted into Park before or after the parking brake is set.
I hope this clarifies this issue!!!