Protein domains shallowly inserting into the membrane matrix are ubiquitous in peripheral membrane proteins involved in various processes of intracellular membrane shaping and remodeling. It has been suggested that these domains sense membrane curvature through their preferable binding to strongly curved membranes, the binding mechanism being mediated by lipid packing defects. Here we make an alternative statement that shallow protein insertions are universal sensors of the intra-membrane stresses existing in the region of the insertion embedding rather than sensors of the curvature per se. We substantiate this proposal computationally by considering different independent ways of the membrane stress generation among which some include changes of the membrane curvature whereas others do not alter the membrane shape. Our computations show that the membrane-binding coefficient of shallow protein insertions is determined by the resultant stress independently of the way this stress has been produced. By contrast, consideration of the correlation between the insertion binding and the membrane curvature demonstrates that the binding coefficient either increases or decreases with curvature depending on the factors leading to the curvature generation. To validate our computational model, we treat quantitatively the experimental results on membrane binding by ALPS1 and ALPS2 motifs of ArfGAP1. 