In scanning probe microscopy, the imaging characteristics in the various interaction channels crucially depend on the chemical termination of the probe tip. Here we analyze the contrast signatures of an oxygen-terminated copper tip with a tetrahedral configuration of the covalently bound terminal O atom. Supported by first-principles calculations we show how this tip termination can be identified by contrast analysis in noncontact atomic force and scanning tunneling microscopy (NC-AFM, STM) on a partially oxidized Cu(110) surface. After controlled tip functionalization by soft indentations of only a few angstroms in an oxide nanodomain, we demonstrate that this tip allows imaging an organic molecule adsorbed on Cu(110) by constant-height NC-AFM in the repulsive force regime, revealing its internal bond structure. In established tip functionalization approaches where, for example, CO or Xe is deliberately picked up from a surface, these probe particles are only weakly bound to the metallic tip, leading to lateral deflections during scanning. Therefore, the contrast mechanism is subject to image distortions, artifacts, and related controversies. In contrast, our simulations for the O-terminated Cu tip show that lateral deflections of the terminating O atom are negligible. This allows a detailed discussion of the fundamental imaging mechanisms in high-resolution NC-AFM experiments. With its structural rigidity, its chemically passivated state, and a high electron density at the apex, we identify the main characteristics of the O-terminated Cu tip, making it a highly attractive complementary probe for the characterization of organic nanostructures on surfaces.