Sure, their mistaken proof of the n->n+1 implication could be taken for what it does prove, and such a theorem could be used in other circumstances to prove different things, but that doesn't really bear on any of the claims in the original context.
Nobody was trying or claiming to be doing any other kind of induction. They said they had the base case and the inductive step, and they were clear about the base case and the inductive step. Their proof of the base case was correct, and their proof of the inductive step did not prove what they said it did. It doesn't matter what it did prove.
However, it must be coupled with a base case >= 2, which isn’t the case here. The only base case proven is 1.
See also “Induction basis other than 0 or 1” [0].
[0] https://en.wikipedia.org/wiki/Mathematical_induction#Inducti...