Suppose $f$ is a modular form of weight $k \ge 2$.

It's "well-known" that there are "periods" $\Omega_-$ and $\Omega_+ \in \mathbb{C}$, such that the $L$-values $L(f, \chi, j)$, for $\chi$ a Dirichlet character and $1 \le j \le k-1$, are $(2\pi i)^j$ times an algebraic multiple of one of $\Omega_{\pm}$.

What is the correct statement of this "well known" result? Colmez claims in his Bourbaki seminar on $p$-adic BSD (here, Section 3.1.3) that

$$L(f, \chi, j) \in \begin{cases} \overline{\mathbb{Q}} \cdot (2\pi i)^j \Omega_+ & \text{if $\chi(-1) = (-1)^j$} \\ \overline{\mathbb{Q}} \cdot (2 \pi i)^j\Omega_- & \text{if $\chi(-1) =-(-1)^{j}$.}\end{cases} $$

On the other hand, Vatsal claims in Theorem 0.1 of this paper (in which he defines canonical choices for the periods $\Omega_{\pm}$ up to $p$-adic units for a given prime $p$) that the conditions should be

$$L(f, \chi, j) \in \begin{cases} \overline{\mathbb{Q}} \cdot (2\pi i)^j \Omega_+ & \text{if $\chi(-1) = 1$} \\ \overline{\mathbb{Q}} \cdot (2 \pi i)^j\Omega_- & \text{if $\chi(-1) =-1$.}\end{cases} $$

Vatsal's statement is repeated verbatim in the MathSciNet review of the paper (here if you have an institutional subscription).

These can't both be right, surely? If $k = 2$ then the only possibility for $j$ is $1$, so the two claims can be reconciled by simply switching the labelling of the two periods; but for $k \ge 3$ then the only way they can both hold is if $\Omega_+ / \Omega_-$ is algebraic, which I gather isn't expected to be the case unless $f$ is CM.

Colmez doesn't give a reference, while Vatsal cites a 1976 paper of Shimura, which I haven't been able to get hold of to check the exact statement.

Which of these two statements are correct? Or are they both correct but I've misunderstood them?