For example, while I certainly don't have the right to demand identification from a random person on the street, I better have the right to demand identification from a police officer giving me an order.
If he's in a firefight and forcefully shoves you out of the way when you walk up asking for his identification, he has no obligation whatsoever to show you his ID. If he is in hot pursuit, you have no business stopping him to ask for his ID. If he has just broken his leg, you can't honestly expect him to identify himself. How many more examples do you want?
I understand the parent is being irrelevantly pedantic, but your comment isn't making it any better.
In the robbing store case, there is probable cause to enter the premises.
But (and in practice this is very similar in effect) there are situations where the law should and does have exceptions for law enforcement officers acting in an official capacity. The obvious one is their privelege to ignore many normal traffic regulations under appropriate circumstances. Similarly, many law enforcement agents are authorized and required to be armed in places where it would be forbidden for anormal citizen.