We introduce new weighted $L^p$-type spaces defined in terms of weight function matrices and characterize the inclusion relations in terms of the defining matrices. Moreover, we provide a detailed study concerning the coincidence with the common (non-weighted) $L^p$-spaces, the (non-)triviality of such weighted spaces and investigate their translation invariance. The obtained results are then applied to particular weight function matrices which are expressed in terms of one single weight function and a positive real parameter. Also variations of this new weighted setting are discussed; more precisely weighted Banach (sub-)spaces of $L^p$ and when weighting the Fourier image of appropriate Banach spaces of functions. The general framework allows to describe the known ultradifferentiable weight function setting by Beurling-Björck which is more original than the approach presented by Braun, Meise and Taylor. When applying the characterization of the inclusion relations to Beurling-Björck-type spaces we are able to emphasize the difference between both ultradifferentiable weight function settings: We construct a technical (counter-)example which is a weight in the sense of Beurling-Björck but violates the standard and crucial convexity condition needed in the Braun-Meise-Taylor setting.