Proper use of "out to lunch", "out for lunch" and "out at lunch"

"Out for lunch" makes me think the person will be bringing the food back with them.

John went out for lunch.

John went out for sandwiches.

John went out for staples.

I have heard the other variations and they seem to mean various things:

John is out at lunch. Can I take a message?

We went out to lunch at the new burger joint.

Can I take you out to lunch?

I have also heard the phrase used to imply someone is "out to lunch" or not entirely there mentally. Context seems to be the only clue that this meaning was intended:

Why did he do that? Is he out to lunch?

When talking to someone while eating, I find this more common:

Can I call you back? I am at lunch.

"Out to lunch" simply means going out (of your place of residence) for the purpose of having lunch. Here, the word lunch is used in the verb form as opposed to its noun form which indeed is a shortened version of luncheon.

Here (central Scotland) we would never use "Out To Lunch" when talking about lunch - it tends to be used to mean that you are mental:-)

You could say going "Out for Lunch" and that would be fine.