Check out Manin's paper Periods of parabolic forms and $p$-adic Hecke series. He only deals with level 1 and even weight, but this is enough to know that Vatsal's statement is not the right one (though not enough to fully confirm Colmez's statement, though it's true as given in Theorem 1 of Shimura's paper you can't get your hands on; you should also be able to dig it out of Mazur–Tate–Teitelbaum or Emerton–Pollack–Weston, which generalizes Vatsal). In particular, see section 1.2: The Periods Theorem of Manin's paper.

Update: In fact, the statement in the form you're looking for is written down in Theorem 1(ii) of Shimura's On the periods of modular forms (doi:10.1007/BF01391466), where he even tells you what you can take as $\Omega_\pm$.

Check out Manin's paper Periods of parabolic forms and $p$-adic Hecke series. He only deals with level 1 and even weight, but this is enough to know that Vatsal's statement is not the right one (though not enough to fully confirm Colmez's statement, though it's true as given in Theorem 1 of Shimura's paper you can't get your hands on; you should also be able to dig it out of Mazur–Tate–Teitelbaum or Emerton–Pollack–Weston, which generalizes Vatsal). In particular, see section 1.2: The Periods Theorem of Manin's paper.

Update: In fact, the statement in the form you're looking for is written down in Theorem 1(ii) of Shimura's On the periods of modular forms (doi:10.1007/BF01391466), where he even tells you what you can take as $\Omega_\pm$ (and the result in the weight 2 case in Shimura's aforementioned 1976 paper was still conditional on a result proved in this 1977 paper).

Check out Manin's paper Periods of parabolic forms and $p$-adic Hecke series. He only deals with level 1 and even weight, but this is enough to know that Vatsal's statement is not the right one (though not enough to fully confirm Colmez's statement, though it's true as given in Theorem 1 of Shimura's paper you can't get your hands on; you should also be able to dig it out of Mazur–Tate–Teitelbaum or Emerton–Pollack–Weston, which generalizes Vatsal). In particular, see section 1.2: The Periods Theorem of Manin's paper.

Check out Manin's paper Periods of parabolic forms and $p$-adic Hecke series. He only deals with level 1 and even weight, but this is enough to know that Vatsal's statement is not the right one (though not enough to fully confirm Colmez's statement, though it's true as given in Theorem 1 of Shimura's paper you can't get your hands on; you should also be able to dig it out of Mazur–Tate–Teitelbaum or Emerton–Pollack–Weston, which generalizes Vatsal). In particular, see section 1.2: The Periods Theorem of Manin's paper.

Update: In fact, the statement in the form you're looking for is written down in Theorem 1(ii) of Shimura's On the periods of modular forms (doi:10.1007/BF01391466), where he even tells you what you can take as $\Omega_\pm$.