Are braid links proper links? Or are the concepts involved unrelated?
According to the definitions in your comment, the closure of the 2 stranded braid with braid word $\sigma_1^6$ is not proper, since the closure is a 2 component link with linking number 3.
It's hard to think of a more straightforward definition than what murakami says, but if you want examples, any link with all pairwise linking numbers even is proper. If you want odd linking number consider three fibers of the Hopf fibration.