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The notion of self-acceleration has been introduced as a convenient way totheoretically distinguish cosmological models in which acceleration is due tomodified gravity from those in which it is due to the properties of matter orfields. In this paper we review the concept of self-acceleration as given, forexample, by [1], and highlight two problems. First, that it applies only touniversal couplings, and second, that it is too narrow, i.e. it excludes modelsin which the acceleration can be shown to be induced by a genuine modificationof gravity, for instance coupled dark energy with a universal coupling, theHu-Sawicki f(R) model or, in the context of inflation, the Starobinski model.We then propose two new, more general, concepts in its place:force-acceleration and field-acceleration, which are also applicable inpresence of non universal cosmologies. We illustrate their concrete applicationwith two examples, among the modified gravity classes which are still inagreement with current data, i.e. f(R) models and coupled dark energy.

Beyond self-acceleration: force- and fluid-acceleration