We present the results of the first spectroscopic observations of two planetary nebula (PN) candidates in the Local Group dwarf irregular galaxy IC 10. Using several spectral classification diagrams, we show that the brightest PN candidate (PN 7) is not a PN, but rather a compact H ii region consisting of two components with low electron number densities. After the rejection of this PN candidate, the IC 10 PN luminosity function cut-off becomes very close to the standard value. With the compiled spectroscopic data for a large number of extragalactic PNe, we analyse a series of diagnostic diagrams to generate quantitative criteria for separating PNe from unresolved H ii regions. We show that, with the help of the diagnostic diagrams and the derived set of criteria, PNe can be distinguished from H ii regions with an efficiency of ∼99.6 per cent. With the obtained spectroscopic data, we confirm that another, 1.7 mag fainter PN candidate (PN 9) is a genuine PN. We argue that, based on all currently available PNe data, IC 10 is located at a distance of 725 +63 -33 kpc [distance modulus (m - M) = 24.30 +0.18 -0.10].