You may check that for $\alpha\leq 0$ when the integral is UV-divergent, the result for $I(n,\alpha)$ you wrote down – which is an analytic function i.e. a result of analytic continuation (and yes, the ignoring of the surface terms for all values of the exponents and dimensions is a way to easily cancel the seemingly divergent but morally vanishing terms which is enough for the analytic continuation) – is still perfectly finite because the pole from $\Gamma(\alpha-n/2)$ (which exists assuming that the argument is a non-positive integer) is cancelled against the pole in the denominator $\Gamma(\alpha)$.
For $n/2=2$, the only values of $\alpha$ for which the pole in the numerator exists and isn't cancelled are $\alpha=2$ which is the logarithmic divergence; and $\alpha=1$ which is the quadratic divergence. Moreover, the pole-like result in this $\alpha=1$ case may be attributed to the "log divergence" part of the integral while the "quadratic divergence" part of the integral, when the integral is properly divided, is equal to zero in dim reg. Dimensional regularization automatically "annihilates" higher-order divergences such as the quartic one (where the admixture of the log divergence is already zero in your integral); a quartic divergence appears in vacuum energy (cosmological constant) diagrams.
Even when the quartic divergences (or the purified quadratic divergences) are rendered as finite by dim reg, it is still true that the theory is sensitive on the parameters of the theory we use at high energies so the elimination of these divergences doesn't mean that we eliminated the finite parameters that still have to be specified and whose values aren't uniquely dictated by any principle if we interpret our QFT as an effective one only.
Maybe the first professor doesn't like the fact that dim reg completely annihilates the infinite term for higher-order power-law divergences – that's a situation he doesn't call a "successful regularization" because in the physical limit, he still wants the result to look infinite. But there's nothing wrong if a regularization assigns a finite value to a naively divergent integral. And indeed, dim reg likes to assign a finite (or even vanishing) result to many diagrams that are divergent in other regularizations. For example, dim reg may automatically preserve the gauge symmetry which means that it cancels the infinite corrections to the mass of the photon.
Analogously, the zeta-function regularization used in 2-dimensional CFTs (and string theory, among other places) automatically preserves the conformal symmetry. The conformal symmetry implies, among other things, that the only symmetry-preserving finite value of the sum of positive integers is $1+2+3+\dots = -1/12$ so similar sums are automatically rendered finite and the value is completely unambiguous in the zeta-function regularization.
The definite answer to your question is:
There is no mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory.
Instead, there are various regularization schemes with their advantages and disatvantages.
Maybe you'll find Chapter B5: Divergences and renormalization of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html illuminating. There I try to abstract the common features and explain in general terms what is needed to make renormalization work. The general belief is that the details of the regularization scheme don't matter, though in fact it is known that sometimes some regularization schemes give apparently incorrect results.
This is to be expected since the unregularized theory is ill-defined, and can be made well-defined in different ways, just as a divergent infinite series can be given infinitely many different meanings depending how you group the terms to sum them up.
If at any time in the future there will be a positive answer to your question, it will be most likely only when someone found a logically sound nonperturbative definition of the class of renormalizable quantum field theories.
On the other hand, if you want to have a mathematically rigorous treatment of some particular regularization schemes for some particular theories, you should read the books by (i) Salmhofer, Renormalization: an introduction, Springer 1999,
and (ii) Scharf, Finite quantum electrodynamics: the causal approach, Springer 1995.
Best Answer
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:
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?