Is there someone that could explain the 2 different representations of W&B sheets to me?
The first one is often found in the POH of an aircraft. In this case a Piper Archer. The second one is actually the same weight and balance, but in stead of a x axel with a distance to the datum, it represents a moment.
Looking at the first image I find it curious to see that the forward limit of 82 inches bends to the left giving the impression that the forward limit is further to the left at a TOM of 2050 lbs. But it's not further left, only the 82 inch line is bending to to the left...
On the second image it is a straight line.
What is the reason behind this?
The second shows a more 'square' envelop.
They are no more than two different ways of doing exactly the same thing.
Some people like one and hate the other, and vice versa :)
I like the bottom one, not least because most electronic devices that offer W&B use that presentation.
Still I think it's weird that a lot of manuals present the 2nd option. You might think that the limits of the envelope are moved further forward or aft with different loading.
When you look at those two formats, they really are very similar to each other, other than the lines in the Piper version not being parallel. The numbers vary, but the format is the same.
There are many different ways of conveying the C of G limitations, but it really boils down to what the Type Certificate Data Sheet says. In the case of the PA-28-161 "Warrior II" (TCDS 2A13):
"Center of Gravity Range
Normal Category (+83.0) to (+93.0) at 1950 lb. or less (+87.0) to (+93.0) at 2325 lb. See NOTE 27.
Utility Category (+83.0) to (+93.0) at 1950 lb. or less (+83.8) to (+93.0) at 2020 lb. Straight line variation between points given."
The manufacturer will provide something pictorial that you can draw on, to assure that your W&B calculation is easy to do, and easy to validate as within limitations. You can use the text limitations from the TCDS to draw your own, they are just saving you the trouble.
If you're moving up to larger aircraft, or helicopters, open your mind, they get to be way more complicated than this!
Sorry, I now see you referenced the Archer, rather than Warrior. The theme will be the same for either plane, just different numbers.