Generay way to find Stability of system
Your first method assumes that the system is entirely characterized by its response to the input $x(t)=\delta(t)$. This would only be the case if the system were linear and time-invariant (LTI). In that case, the input-output relation would have the form

$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\tag{1}$$

where $h(t)$ is the system's impulse response. However, the given input-output relation is not of the form $(1)$, and hence the system is not LTI. Consequently, the system's response to $x(t)=\delta(t)$ does not say much about the system's behavior in general. That's why your first method of determining the stability is not valid.

The second method is valid. You found a bounded input signal for which the output is unbounded, and in this way you've shown that the system is not BIBO-stable.