I am a bit lost in the concepts of stability theory. Consider the (non-linear) ODE $x' = \varphi(x)$ in some Banach space with a unique stationary point $x_*.$
Then we could say that the fixed point is (globally) exponentially stable if there exists an $\alpha > 0$ such that
$$ \|x_* - x(t)\| \leq e^{-\alpha t} \|x_* - x(0)\| $$ holds for any $t \geq 0$
and any initial $x(0).$ But we could also define the concept of stability through the solution curves. That is, the system is exponentially stable if there is an $\alpha > 0$ such that
$$ \|x_1(t) - x_2(t)\| \leq e^{-\alpha t} \|x_1(0) - x_2(0)\| $$ holds for any $t \geq 0$ and $x(0), y(0).$
The second definition readily implies the first property of $x_*.$ But I do not see any simple proof of the converse statement and do not have any counterexample either. Still, I naively think that these two stabilities are the same. Please, could you give any reference to these types of results?
The requirement for exponential stability is a little less strict as your first inequality. Namely, one can also include a constant positive but finite factor $\beta$, such that initial transients are allowed
$$ \|x_* - x(t)\| \leq \beta\,e^{-\alpha t} \|x_* - x(0)\|\quad \forall\ t \geq 0. $$
For your proposed second definition I initially though that it would still allow for limit cycles. However, there is no driving force, which causes the two solutions to synchronize. One other possibility is that the system is unstable and the state goes to infinity but at a slower and slower rate. I have tried to come up with an example of this, but have not been able to proof that neighboring solutions of the system below exponentially converge
$$ \dot{x} = x\,e^{-x^4}, $$
but numerically simulations do seem to agree with this.
For nonautonomous systems one can come up with counter examples, where there are bounded state trajectories that have a basin of attraction, such as
\begin{align} r(t) &= 2 + \sin(\omega\,t), \\ w(t,x) &= (1 + 10^8\,(r(t) - x)^8)^{-1}, \\ \dot{x} &= (w(t,x) - 1)\,x + w(t,x) \left(k\,(r(t) - x) + \dot{r}(t)\right). \end{align}