What exactly is Cramer-Wold device? (What's the difference b/w Cramer-Wold theorem and Cramer-Wold device?)

Question

I'm a bit confused about the exact statement of the Cramer-Wold device. I've read multiple sources which seem to be giving different definitions (the first one is more common).

1. Random vectors $X_n \in \mathbb{R}^d$ satisfy $X_n \rightarrow_d X$ if and only if $\langle a, X_n \rangle \rightarrow_d \langle a, X \rangle$ for all $a \in \mathbb{R}^d$.

(Source: Proposition 2.7)

2. If $\langle a, X_n \rangle \rightarrow_d \langle a, X \rangle$ for all $a \in \mathbb{R}^d$ with $\|a\| = 1$, then $X_n \rightarrow_d X$.

(Source: Theorem 13)

Also, What's the difference between Cramer-Wold theorem and Cramer-Wold device? Is there a difference?