I realize that no one addressed the question of the OP about Calabi-Yau manifolds. Two remarks:

1. The equality $\mathrm{Bir}(X)=\mathrm{Aut}(X)$ holds if $K_X$ is nef and $\mathrm{Pic}(X)=\mathbb{Z}$ (see for instance the introduction of this paper). This applies in particular to Calabi-Yau complete intersections (of dimension $\geq 3$).

2. There are examples of birational automorphisms of holomorphic symplectic manifolds which are not biregular, see §6 of this paper.

I realize that no one addressed the question of the OP about Calabi–Yau manifolds. Two remarks:

1. The equality $\mathrm{Bir}(X)=\mathrm{Aut}(X)$ holds if $K_X$ is nef and $\mathrm{Pic}(X)=\mathbb{Z}$ (see for instance the introduction of Chen - Rational self maps of Calabi–Yau manifolds). This applies in particular to Calabi–Yau complete intersections (of dimension $\geq 3$).

2. There are examples of birational automorphisms of holomorphic symplectic manifolds which are not biregular, see §6 of Beauville - Some remarks on Kähler manifolds with $c_1 = 0$.