The reactions γp → ηp and γp → η 0 p are measured from their thresholds up to the center-of-mass energy W ¼ 1.96 GeV with the tagged-photon facilities at the Mainz Microtron, MAMI. Differential cross sections are obtained with unprecedented statistical accuracy, providing fine energy binning and full production-angle coverage. A strong cusp is observed in the total cross section for η photoproduction at the energies in the vicinity of the η 0 threshold, W ¼ 1896 MeV (E γ ¼ 1447 MeV). Within the framework of a revised ηMAID isobar model, the cusp, in connection with a steep rise of the η 0 total cross section from its threshold, can only be explained by a strong coupling of the poorly known Nð1895Þ1=2 − state to both ηp and η 0 p. Including the new high-accuracy results in the ηMAID fit to available η and η 0 photoproduction data allows the determination of the Nð1895Þ1=2 − properties.