Usage of twists at pairing-based cryptography
I'm not sure I understood your question, but... twists are used to improve performance. In your example, without twists, $G_2$ is $E(\mathbb{F}_{p^{16}})$. With a quartic twist, you can use $E'(\mathbb{F}_{p^{4}})$, which is much more efficient (the reason for this is that the homomorphism $\Psi$ maps the point coordinates to sparse elements of $\mathbb{F}_{p^{16}}$, i.e. with a lot of zero coefficients, which can be multiplied much faster by ignoring the internal multiplications by zero).

The splitting of the scalar $m$ into $m_1, m_2$ is known as "GLV method". While it is used to speed up point multiplications, as far as I know it can't be applied directly to the pairing computation. But it can be used for pairing-based schemes that use point multiplication along with the pairing computation.