I have this definition of a projective complex algebraic surface that is a complete intersection.
A surface $S\subset\mathbb{P}^{r+2}$ is said to be a complete intersection if it is a trasversal intersection of $r$ hypersurfaces $Y_1,\dots,Y_r$ that are smooth at the points of intersections.
My question is: is this surface $S$, defined like this, always smooth?
(OK, reposting my comment to remove this question from the unanswered list.)
Yes: the tangent space to the surface at a point is the intersection of the tangent spaces of the hypersurfaces, and your hypotheses ensure that this intersection has dimension 2 at every point of the surface.