You're totally right. The Wikipedia definition of the renormalization is obsolete i.e. it refers to the interpretation of these techniques that was believed prior to the discovery of the Renormalization Group.
While the computational essence (and results) of the techniques hasn't changed much in some cases, their modern interpretation is very different from the old one. The process of guaranteeing that results are expressed in terms of finite numbers is known as the regularization, not renormalization, and integrating up to a finite cutoff scale only is a simple example of a regularization.
However, the renormalization is an extra step we apply later in which a number of calculated quantities is set equal to their measured (and therefore finite) values. This of course cancels the infinite (calculated) parts of these quantities (I mean parts that were infinite before the regularization) but for renormalizable theories, it cancels the infinite parts of all physically meaningful predictions, too.
However, the renormalization has to be done even in theories where no divergences arise. In that case, it still amounts to a correct (yet nontrivial) mapping between the observed parameters and the "bare" parameters of the theory.
The modern, RG-based interpretation of these issues changes many subtleties. For example, the problem with the non-renormalizable theory is no longer the impossibility to cancel the infinities. The infinities may still be regulated away by a regularization but the real problem is that we introduce an infinite number of undetermined finite parameters during the process. In other words, a non-renormalizable theory becomes unpredictive (infinite input is needed to make it predictive) for all questions near (and above?) its cutoff scale where its generic interactions (higher-order terms) become strongly coupled.