I think the first alternative is better, but it depends on how you define ΔE and Δt. For ## \Delta t ## it is customary to use the lifetime of the decaying particle, and for ## \Delta E ## the full-width-half-maximum (FWHM) of the line in the energy spectrum. The line shape is typically fitted with a
Lorentzian, but the variance of such a Cauchy distribution is infinite! That's why one resorts to using the FWHM measure.
The line shape can be written in the form $$
{1 \over \pi} {{\Gamma / 2} \over {(\omega - \omega_0)^2 + (\Gamma / 2)^2}}
$$ and from this you find for ## \omega - \omega_0 = \pm \Gamma / 2 ##: $$
(\Delta \omega)_\text{FWHM} = \Gamma = 1 / \tau
$$ with ## \tau ## the lifetime of the excited state. Dividing by ## 2 \pi ## to convert from angular to normal frequencies, and multiplying by ## h ## I find $$
\Delta E = {h \over {2 \pi \tau}} = {\hbar \over \tau} \ .
$$
You see, this is only indirectly related to the "usual" uncertainty relations.