Regularization and renormalization are conceptually distinct.
As you essentially indicate, regularization is the process by which one renders divergent quantities finite by introducing a parameter $\Lambda$ such that the "original divergent theory" corresponds to a certain value of that parameter. I put "original divergent theory" in quotations because strictly speaking, the theory is ill-defined before regularization.
Once you regularize your theory, you can calculate any quantity you want in terms of the "bare" quantities appearing in the original lagrangian (such as masses $m$, couplings $\lambda$, etc.) along with the newly introduced regularization parameter $\Lambda$. The bare quantities are not what is measured in experiments. What is measured in experiments are corresponding physical quantities (the physical masses $m_P$, couplings $\lambda_P$, etc.).
Renormalization is the process by which you take the regularized theory, a theory written in terms of bare quantities and the regularization parameter $(\Lambda, m, \lambda, \dots)$, and you apply certain conditions (renormalization conditions) which cause physical quantities you want to compute, such as scattering amplitudes, to depend only on physical quantities $(m_P, \lambda_P, \dots)$, and in performing this procedure on a renormalizable quantum field theory, the dependence on the cutoff disappears. So, in a sense, renormalization can be thought of as more of a procedure for writing your theory in terms of physical quantities than as a procedure for "removing infinities." The removing infinities part is already accomplished through regularization.
Beware that what I have described here is not the whole conceptual story of regularization and renormalization. I'd highly recommend that you try to read about the following topics which give a more complete picture of how renormalization is thought about nowadays:
- effective field theory
- wilsonian renormalization
- renormalization group
- renormalization group flows
You may also find the following physics.SE posts interesting/illuminating:
What exactly is regularization in QFT?
Regulator-scheme-independence in QFT
Why is renormalization necessary in finite theories?
Why do we expect our theories to be independent of cutoffs?