Exponential Stability of State Observers
With the estimation error defined as \$e = x - \hat{x}\$, uniform exponential stability (UES) of the error system \$\dot{e} = (A-LC)e\$ does not imply either \$x \to 0\$ or \$\hat{x} \to 0\$, as you suggested. Actually, the reason that UES is a stronger condition than \$e \to 0\$ is that UES tells us something about how fast \$e\$ converges to \$0\$. Roughly speaking, the word exponential tells us that \$e\$ approaches zero at least as fast as some exponential decay, while uniform tells us that this convergence rate does not change with time. This 'uniformity' is important when dealing with time-varying \$A(t)\$, because if the convergence is not uniform we can have pathological cases where the convergence to \$0\$ becomes slower and slower with time, so that we sustain a large error \$e\$ for an arbitrarily long period of time.

To provide context for others reading this question, in this state estimation problem we want to estimate the state \$x\$ of a (linear, time-varying) state-space system
$$\dot{x} = A(t)x, \qquad y = C(t) x$$
by forming some estimate \$\hat{x}\$. We can form such an estimate by simulating the system and applying some feedback from the output as
$$\dot{\hat{x}} = A(t) \hat{x} + L(t)(y-C(t)\hat{x})$$
and we want to design \$L(t)\$ so that our estimate \$\hat{x}\$ converges to \$x\$ after some time. We analyse this by introducing the error \$e=x-\hat{x}\$, which satisfies
$$\dot{e} = (A(t) - L(t)C(t))e$$
and by analysing the stability of this system we can discover if the error \$e\$ goes to zero or not. 

To make the discussion of stability more precise, if we say that the error system is exponentially stable we mean that for each initial condition \$(t_0, e(0))\$ we have
$$
\| e(t) \| \leq \alpha \| e(0) \| \exp(-\beta t)
$$
for some constants \$\alpha&gt;0\$, \$\beta&gt;0\$. If we say that it is uniformly exponentially stable, we mean that we can find some \$\alpha&gt;0\$, \$\beta&gt;0\$ such that this inequality is satisfied for any initial time \$t_0\$. When dealing with nonlinear systems, we sometimes also say that the stability is either local or global depending on whether we also restrict \$e(0)\$ to be within some region of the state space or not, but for linear systems like the one considered here we don't need to worry about this, as stability is automatically global.