First, it’s notable: You have to realize Einstein’s theory is a model of gravitational phenomenon per our observational affirmations. In reality, it is likely beyond our human brains to understand the forces and dimensions at play. Because of our feeble minds, we are forced to use analogies and models, which would make space-time curved due to distortion of matter-matter attraction (gravitational force). These words are better realized by mathematics via the Einsteinian et alia equations used to model gravitation of matter (as well as massless particles like photons within this space-time*).
But to answer your question less indirectly: Gravity is more a distortion of space-time and not due to it’s curvature autonomously. Gravity’s essence cannot be directly observed, but will indeed be revealed as a curvature of space-time; matter is doing this and matter is susceptible to following this curve. So gravity, in an Einsteinian sense, is matter-induced space-time curvature which is influential to matter. So, gravity itself is both (to use the wording of your question) “the effect from a distortion in the space-time fabric [and] the actual curvature of space-time”.
*Photons are abnormal matter if not non-matter. They are without mass (a defining trait of matter) and move at speeds no matter can achieve. Yet, they are influenced by space-time in a “curvy” manner just like matter. Look up an Einstein Ring and read how that works and you’ll begin to understand a bit of what astronomers and physicists mean by their pop-culturized words.