Uncovered Interest Parity (UIP) condition approximation
From the basic equation you get
\begin{eqnarray*}
\frac{e^E_{t+1}}{e_{t}} &amp; = &amp; \frac{1+i_t}{1+i^*_t} \\
\\
(1+i^*_t) \cdot e^E_{t+1} &amp; = &amp; (1+i_t) \cdot e_t \\
\\
e^E_{t+1} - e_t &amp; = &amp; i_t \cdot e_t - i^*_t \cdot e^E_{t+1}.
\end{eqnarray*}
Now comes the approximation. You divide either by $e_t$ or by $e_{t+1}^E$.

First method
$$
\frac{e^E_{t+1} - e_t}{e_t} = i_t - i^*_t \cdot \frac{e^E_{t+1}}{e_t} \approx
i_t - i^*_t.
$$
Second method
$$
\frac{e^E_{t+1} - e_t}{e^E_{t+1}} = i_t \cdot \frac{e_t}{e^E_{t+1}} - i^*_t  \approx
i_t - i^*_t.
$$
I would say there is no crucial difference, both of these just assume that the proportional change in the exchange rate multiplied by the interest rate is an order of magnitude lower than either the interest rate or the change in the exchange rate alone.