DHKE choice of private keys In some literature it is written that the private key should be chosen random from 

{2,3, ..., p-1} 

and in some it says

{2,3, ..., p-2}

I guess p-1 is a weak private key, but why is that so?
The calculation would be:

A = k pub, A = αp-1 mod p
$p-1$ is certainly a weak private key, in that it forces the shared secret to be a constant value as:

$a^{p-1} \bmod p = 1$

(This is known as Fermat's Little Theorem; no relation to Fermat's Last Theorem)

Hence, if either side uses $p-1$ as there private key, then both sides would arrive at the value of 1, and that's not great for security.