The comprehensive description of both the electrical transport along conductive domain walls (CDWs) in lithium niobate (LNO) single crystals and the charge injection at the interfacing metal electrodes, emerged to be a complex challenge. Recently, a heuristic evaluation allowed to postulate the "R2D2" equivalent-circuit model (consisting of two parallel resistor-diode pairs) to appropriately match the DC current-voltage (I-V) characteristics. Here, we carefully revisit the interfacial electrical behavior, i.e., the diode part of the equivalent circuit model, since many more processes beyond the diode-related electron hopping transport (HT) assumed so far, may concurrently occur, such as thermionic emission (TE), Fowler-Nordheim tunneling (FNT), space-charge limited conduction (SCLC), and others more. The "R2D2" model thus needs to be generalized into an "R2X2" circuit model (with X = HT, TE, FNT, and others) to fit to the experimental data. Moreover, to double check for the best I-V curve fitting to the different theories, we apply a higher-harmonic DW current-contribution (HHCC) analysis, i.e., an AC I-V inspection, that allows us to discriminate between all these possible models with much higher precision than from pure DC I-V curve fitting. Both the AC and DC analysis reveal well consistent results, finally finding that the FNT model accounts best for the domain-wall/electrode junctions investigated here.