Elastic properties of biological membranes are involved in a large number of membrane functionalities and activities. Conventionally characterized in terms of Young's modulus, bending stiffness and stretching modulus, membrane mechanics can be assessed at high lateral resolution by means of atomic force microscopy (AFM). Here we show that the mechanical response of biomimetic model systems such as supported lipid bilayers (SLBs) is highly affected by the size of the AFM tip employed as a membrane indenter. Our study is focused on phase-separated fluid-gel lipid membranes at room temperature. In a small tip radius regime (approximate to 2 nm) and in the case of fluid phase membranes, we show that the tip can penetrate through the membrane minimizing molecular vertical compression and in absence of molecular membrane rupture. In this case, AFM indentation experiments cannot assess the vertical membrane Young's modulus. In agreement with the data reported in the literature, in the case of larger indenters (>2 nm) SLBs can be compressed leading to an evaluation of Young's modulus and membrane maximal withstanding force before rupture. We show that such force increases with the indenter in agreement with the existing theoretical frame. Finally, we demonstrate that the latter has no influence on the number of molecules involved in the rupture process that is observed to be constant and rather dependent on the indenter chemical composition.