The income effect is defined as $\mathbf{-\frac{\partial x_i}{\partial m}x_i}$.

Let $x_i$ be a normal good; that is, a good whose Marshallian demand increases with an increase in income $\left(\frac{\partial x_i}{\partial m} > 0\right)$. Even if there was no substitution effect $\left(\frac{\partial h_i}{\partial p_i} = 0\right)$ from an increase in own-price, the amount of $x_i$ consumed would still be reduced as the consumer is effectively poorer. Hence the net income effect in this case is negative, which necessitates the negative sign since $x_i$ is non-negative. Analogous logic holds if $x_i$ is an inferior good instead.

The income effect is defined as $\mathbf{-\frac{\partial x_i}{\partial m}x_i}$.

Let $x_i$ be a normal good; that is, a good whose Marshallian demand increases with an increase in income $\left(\frac{\partial x_i}{\partial m} > 0\right)$. Even if there was no substitution effect $\left(\frac{\partial h_i}{\partial p_i} = 0\right)$ from an increase in own-price, the amount of $x_i$ consumed would still be reduced as the consumer is effectively poorer. Hence the net income effect in this case is negative, which necessitates the negative sign since $x_i$ is non-negative. Analogous logic holds if $x_i$ is an inferior good instead.

This interpretation is consistent with Nicholson & Snyder (p. 156).