The forces acting on the bob are Tension of the string and Force of Gravity, which can be decomposed into two components: Normal or Radial one, which is antiparallel to Tension (mgcosθ) and the Tangential one, perpendicular to Tension (mgsinθ).
All authors agree that Tension cancels the Radial Component of Gravity and that the restoring force is the Tangential Component of Gravity, but they differ in that:
a) For example, text books like Giancolli, Serway or Halliday, paint Tension as having exactly the same size as the Radial Component of Gravity (see pictures here), as if T = mgcosθ, as if neutralizing radial acceleration were enough to ensure circular motion.
b) For others, like in this picture of Young, Freedman and Ford, this animation of Wikipedia or both the pictures and statements of the site physicsclassroom, Tension is always greater than the Radial Component of Gravity. For them, T = mgcosθ + mv2/r, the term mv2/r being the one that provides the centripetal acceleration justifying that the bob moves in a circle (otherwise it would not).
An argument in favor of a) seems to be the nature of Tension Force: Tension is analogous to Normal Force and just like Normal Force adapts its magnitude to counter-act Weight, shouldn’t Tension do the same here and just neutralize the component of Gravity along the line of the string? Can’t we say that Tension acts as centripetal force just by depriving the bob of radial acceleration, without the need of a second term?
So the questions are:
1/ What is the right formula, (a) T = mgcosθ or (b) T = mgcosθ + mv2/r?
2/ Is really Tension like Normal Force and if so how does that circumstance play here?
After some reflection, I have thought of my own answer, which I will provide for comments.