Given a control system $\dot x = Ax + Bu$ and $y = Cx$. Suppose we use state feedback to create $u = +Kx$ where $K$ is the gain matrix.
Subbing into above equation, we have $\dot x = Ax + Bu = Ax + BKx = (A+BK)x$
In the literature $(A+BK)$ is frequently referred to as the closed loop system. But recall that in classical control we have $\dfrac{G(s)}{1+H(s)G(s)}$ referred to as the closed loop system. So we literally have two definitions and I cannot see if they are compatible or not.
Are the two definitions compatible? What is the definitive reason that $(A+BK)$ is called the closed loop system?
There are two major notions in control theory: an open loop system and a closed loop one. The open loop means that you consider a system as an input-output function $$ y=Gv. $$ One can plug in different input functions $v$ and see what happens with the output signal. The input is often split into the control input $u$ (the one you can choose) and the disturbance input $w$ (the one you cannot choose or maybe even know), so the systems splits as (I will talk about linear systems only) $$ y=G_1u+G_2w. $$ The closed loop system means that you have another system that you connect to the original one. This another system is called a feedback. It gives you a dependence how the control input is formed using the signals we know, that is $y$ $$ u=Hy. $$ It may though happen that the control circuit is affected by some noise, so let us consider a feedback with noise $$ u=Hy+\xi. $$ So what is the closed loop system? You just need to plug the feedback system into the open loop system like this $$ y=G_1(Hy+\xi)+G_2w $$ and solve the whole thing with respect to $y$ $$ y-G_1Hy=G_1\xi+G_2w\qquad\Rightarrow\qquad y=(I-G_1H)^{-1}G_1\xi+(I-G_1H)^{-1}G_2 w. $$ The closed loop system may have many different forms, depending on what one takes for the open loop, what one takes for disturbances/noises, maybe some are zeros etc.
In your first example, the open loop is $\dot x=Ax+Bu$, where $u$ is the control and no disturbance, and the feedback is $u=Kx$, no noise. Here $A+BK$ is called the system matrix of the closed loop system. It is not the closed loop system, the closed loop system is the differential equation with the matrix $A+BK$. In the second example, you have the open loop $y=Gu$ and the feedback $u=-Hy+\xi$, so the closed loop becomes $$ y=G(-Hu+\xi)=-GHy+G\xi\qquad\Rightarrow\qquad y=\frac{G}{1+GH}\xi. $$ So the closed loop is the resulting system that you get after having chosen and plugged in a particular control feedback strategy. It shows the dependence of the output only on the disturbances, the effect that it is often desirable to minimize.
UPDATE: When the linear system is considered in the time domain, one works with the state space representation of the linear differential equation where all functions depend on time $$ G\colon\ \left\{ \begin{array}{lll} \dot x&=&Ax+Bu,\\ y&=&Cx+Du. \end{array} \right. $$ The feedback is, in general, also given by a differential equation $$ H\colon\ \left\{ \begin{array}{lll} \dot q&=&A_fq+B_fy,\\ u&=&C_fq+D_fy. \end{array} \right. $$ The situation in the first example when $y=x$ (the full information case $C=I$, $D=0$) and $u=Kx$ (the static feedback $D_f=K$, others are zero) is the simplest case.
Alternatively, one can make the Laplace transform to get the multiplication form of the operators $G$ and $H$ in the frequency domain as $$ y(s)=\underbrace{[C(sI-A)^{-1}B+D]}_{G(s)}u(s),\qquad u(s)=\underbrace{[C_f(sI-A_f)^{-1}B_f+D_f]}_{=H(s)}y(s). $$