There are two ways to see why {1}:Delta[0]-->Delta[1] is not inner anodyne: inner anodyne maps are (a) bijective on 0-simplices and (b) acyclic cofibrations in the Joyal model structure, i.e., monic weak categorical equivalences. One has to be a bit careful with these things, because it turns out there are maps satisfying (a) and (b) which are not inner anodyne. Alexander has a short paper which explains a simple counterexample: https://arxiv.org/abs/1904.04965
But yeah, the map {1}:Delta[0]-->Delta[1] satisfies neither (a) nor (b), so is not an inner anodyne map.