pischei chochmah:
He also said that tehillim is not talmud torah but is a form of davening (because, among other reasons, talmud Torah involoves two mitzvas: knowing & ameilus, neither of which is likely to happen when one says tehillim. With the exception of Mishnayos, that has a mystical power to simply saying the words, all Torah shbaal peh is meaningless without knowing the content.
I’m moderately sure that tehillim is Torah she’bichsav not Torah sheba’al peh and that all Torah she’bichsav falls into the same category that you describe about mishnayos.
Also,I think I’ve heard that tehillim is both- but lacking a source, I have no proof.