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.
The renormalization condition is exactly that the propagator has a pole and residue of 1 at
\begin{equation}
\gamma^\mu p_\mu=M
\end{equation}
which leads to
\begin{equation}
p^2=(\gamma p)^2=M^2
\end{equation}
I think your problem here is that you mermorized the renormalization condition as $p^2=-M^2$ which is not true. To understand the condition properly, just bring up the free-field propagator
\begin{equation}
\frac{i}{p^2-M^2}
\end{equation}
or
\begin{equation}
\frac{i}{\gamma^\mu p_\mu-M}
\end{equation}
notice the real-corrected propagator behaves the same as the free-field near the pole, in this way you may not have a memoral mistake.
Best Answer
If there is a soliton background (of dashed lines), then this tadpole diagram with extended legs like
shall not be zero due to Jackiw-Rebbi or Goldstone-Wilczek effect; the soliton background (dashed lines) will induce nontrivial (fractionalized) induced quantum numbers (or simply the charge or vacuum expectation value) for the scalar bosons (solid line); even if the original scalar bosons(solid line) are massless.
For example. see the bosonic anomaly in arxiv-1403.5256 - Bosonic Anomaly, Induced Fractional Quantum Number and Degenerate Zero Modes.
Interestingly, such an effect takes place on the surface of Symmetry Protected Topological(SPT) States as noticed in this Ref. And see Reference therein, e.g. about J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981) and R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).