A couple of points here.
First of all, there is no investigation or paperwork following an emergency. There is paperwork following an accident - but the paperwork will be the same whether you declare an emergency or not. Unless you avoid the accident by asking for help, of course, in which case the paperwork requirement goes away!
In your 5 scenarios, case 5 is, IMHO, different to the others. I'm assuming that what you mean is that there was nothing wrong, except that the pilot c0cked up the landing and there was some damage? Not much point declaring an emergency in this case - the damage is already done before you've had a chance!
As for the others, if you're short of fuel, or short of options, or there's a mechanical problem, then it's either Mayday or Pan Pan, depending how urgent the problem is. Simple.
I'm interested that in two of your options you mention weather problems. Weather problems don't automatically create an emergency situation.
Example: you take off and fly towards a cold front. You hope to reach your destination before you reach the start of the frontal weather. About 20 miles from your destination, you encounter heavy rain, and turn around. Is this an emergency? No - IMHO, if the weather behind you is clear, and you have plenty of fuel left, then you have plenty of options, so no need to bother anyone else. Of course, that doesn't mean you shouldn't tell anyone - I'd tell anyone I was talking to. If I was receiving a LARS or FIS, I'd tell the controller I was diverting. On contacting my diversion field, I'd tell them I'm a (for example) "PA28, from A to B, diverting to you due weather". This at least explains why you're changing course, why you haven't phoned for PPR, why you maybe don't have the airfield diagram to hand - but there's no need for special priority, no need for a fire truck to meet you, and no need to declare an emergency.
Example 2: you take off and stay below the overcast layer of clouds. You cross a range of hills, and then encounter bad weather ahead. You turn around, only to find that the clouds bases have lowered, and you can't get back across the range of hills that you passed earlier. This quite possibly is an emergency. Right now, I want all the help I can get. I want to know if there's a farm-strip that's not shown on my chart, or if another pilot has just reported a gap in the clouds a few miles away. If I'm running low on fuel or if the clouds are getting lower, then I will have to land somewhere very soon, even if it's off airport. With the cloud-base lowering, the chances are I'm not going to want to hang around extending downwind in a 1000' circuit while some twin is 4 miles out on the ILS - I want to be on the ground, and I want to be on the ground asap.
At least that's my personal interpretation - don't know if everyone else will agree!
FFF
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