You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least the following properties

K-theory of an affine scheme $\mathrm{Spec}\,R$ is given by the algebraic K-theory of projective $R$-modules in the sense of Quillen

K-theory satisfies Zariski descent (as a spectrum)

Since every scheme has a Zariski cover by affine schemes it is clear that there's at most one definition that satisfies the above two properties. At least when $X$ is quasi compact quasi separated this can be expressed more geometrically as the algebraic K-theory of *perfect complexes*. An introduction that does not require much background can be found in the classical paper by Thomason and Trobaugh, *Higher Algebraic K-theory of schemes and of Derived categories* (although if you want to invest some time learning some modern homotopy theory can only be beneficial -- many of the proofs in Thomason-Trobaugh can be simplified if you have more modern technology at your disposal).

Similarly for G-theory the "correct" analogue for qcqs schemes is the algebraic K-theory spectrum of bounded pseudocoherent complexes (also known as "almost perfect" complexes in parts of the literature).

Both notions of K-theory and G-theory recover Quillen's definition when working on Noetherian schemes (and for K-theory in a much greater generality -- whenever the scheme has an ample family of line bundles).

1more comment