question on formulations of Generalized Continuum Hypothesis and Singular Cardinal Hypothesis I hope this is not a silly question(well, not too silly, I hope). After all,  a relevent question at a deeper level is already out there, even though it seems the solution is missing.

Why not formulate SCH as simply, $\kappa^{\operatorname{cf}{\kappa}}=\kappa^{+}$?
Why not formulate GCH as, if $2^{\operatorname{cf}{\kappa}}<\kappa$, then $2^\kappa=\kappa^{+}$?
Or both?
Is it merely a historical coincidence?
My naive guess is that GCH was proposed prior to SCH by coincidence in some sense, set theorists prefer to propose hypotheses that can be linear-ordered in strength.
 A: First of all, $\mathsf{GCH}$ stands for "Generalized Continuum Hypothesis". 
$\mathsf{SCH}$ is not what you say. Rather, it is the statement that, for singular cardinals, $\kappa^{\rm{cf}(\kappa)}$ is "as small as possible". The usual formulation, 
 $$ \kappa^{\rm{cf}(\kappa)}=2^{\rm{cf}(\kappa)}+\kappa^+, $$ 
achieves this: First of all, it says nothing about regular cardinals $\kappa$, because ${\rm cf}(\kappa)=\kappa$ gives us that $2^{\rm{cf}(\kappa)}+\kappa^+=2^\kappa+\kappa^+=2^\kappa$, by Cantor. And $\kappa^{\rm{cf}(\kappa)}=\kappa^\kappa=2^\kappa$, by Schröder-Bernstein. So $\kappa^{\rm{cf}(\kappa)}=2^{\rm{cf}(\kappa)}+\kappa^+$ holds in this case. 
Second, for singular $\kappa$, $\kappa^{\rm{cf}(\kappa)}\ge\kappa^+$, by König, and $\kappa^{\rm{cf}(\kappa)}\ge 2^{\rm{cf}(\kappa)}$, by monotonicity. So $\kappa^{\rm{cf}(\kappa)}\ge 2^{\rm{cf}(\kappa)}+\kappa^+$, and $\mathsf{SCH}$ is simply the statement that this lower bound is actually achieved. 
The statement you propose, $\kappa^{\rm{cf}(\kappa)}=\kappa^+$, is too strong, and in fact it is equivalent to $\mathsf{GCH}$: It clearly implies $2^\kappa=\kappa^+$ for $\kappa$ regular. But this gives us $\mathsf{GCH}$ by an inductive argument: If it holds for all cardinals below the singular cardinal $\kappa$, then $\kappa$ is actually strong limit, and $\kappa^{\rm{cf}(\kappa)}=2^\kappa$, by Bukovský-Hechler. But then your statement gives us $2^\kappa=\kappa^+$, and $\mathsf{GCH}$ follows.
(Conversely, if $\mathsf{GCH}$ holds, then Bukovský-Hechler gives us that $\kappa^{\rm{cf}(\kappa)}=\kappa^+$ for $\kappa$ singular, so your statement holds, since it is always true for $\kappa$ regular.)
However, $\mathsf{GCH}$ is too restrictive as a substitute for $\mathsf{SCH}$. The point of $\mathsf{SCH}$ is that, even if we allow the exponential function to behave arbitrarily on regular cardinals (as shown consistent through Easton's result), the behavior at singular is still completely determined. Silver's work showed that this is not an unreasonable assumption, and that in fact, if it is not true, then the first counterexample must be a $\kappa$ of cofinality $\omega$. Jensen's work on $0^\sharp$ goes further, and tells us that if we start with a "small" universe, such as $L$, then forcing cannot produce violations to $\mathsf{SCH}$, even though we can make $\mathsf{GCH}$ fail pretty drastically.
Your reformulation does not give us any of these insights, since it is just $\mathsf{GCH}$.

On the other hand, your reformulation of $\mathsf{GCH}$ is too weak, since it says nothing about regular cardinals, and does not even imply $2^\kappa=\kappa^+$ at singulars since, for example, $2^{\aleph_0}$ could be much larger than $\aleph_{\omega+1}$, which of course implies that $2^{\aleph_\omega}>\aleph_{\omega+1}$ as well.
Nevertheless, if $2^{\rm{cf}(\kappa)}\ge\kappa$, then in fact the inequality is strict (by König) and moreover $\kappa^{\rm{cf}(\kappa)}\le 2^{\rm{cf}(\kappa)}$. Since the other inequality is trivial, we get that $\kappa^{\rm{cf}(\kappa)}= 2^{\rm{cf}(\kappa)}$ in this case. Hence, your reformulation of $\mathsf{GCH}$ is actually equivalent to $\mathsf{SCH}$.

Let me close by mentioning that though $\mathsf{GCH}$ and $\mathsf{SCH}$ serve different purposes, they are actually closely related. There is a caveat here, as the first says that the exponential function is as small as possible everywhere, a statement easily falsifiable by forcing, while the second says that $\kappa^{\rm{cf}(\kappa)}$ is completely determined at singular cardinals, regardless of the behavior of the exponential function at regular cardinals. Its negation is consistent as well, but this requires (a modest amount of) large cardinal strength (the precise calibration of which is due to Gitik). 
Of course the two statement are related, since both say something about the exponential being small. In a sense, $\mathsf{SCH}$ can be seen as an attempt to formulate a natural conjecture in the light of Easton's theorem, namely that "a trace of $\mathsf{GCH}$" is nonetheless true. Now we know that this needs not be the case, and $\mathsf{GCH}$ and $\mathsf{SCH}$ can fail everywhere. 
[The emphasis on $\kappa^{\rm{cf}(\kappa)}$ and $2^\kappa$ rather than the more general $\kappa^\lambda$ is really not important, since as a consequence of Bukovský-Hechler, the values of this two-variable function are completely determined by these two functions. More precisely, we have:

Theorem. Let $\kappa$ and $\lambda$ be infinite cardinals. Let $\tau=\sup_{\rho<\kappa}|\rho|^\lambda$. Then
   $$ \kappa^\lambda=\left\{\begin{array}{cl} 
2^\lambda & \mbox{if }\kappa\le 2^\lambda,\\ 
\kappa\cdot\tau & \mbox{if }\lambda<\rm{cf}(\kappa),\\ 
\tau & \begin{array}{l}\mbox{if }\rm{cf}(\kappa)\le\lambda,2^\lambda<\kappa,\mbox{ and }
\\ \rho\mapsto|\rho|^\lambda\mbox{ is eventually constant below }\kappa,\end{array}\\ \kappa^{\rm{cf}(\kappa)} & \mbox{otherwise.}\end{array}\right. $$

For details, see for example these notes of mine.]
