Of course you can't rigorously prove something that's not true.

Your hindsight is accurate, but more than just recognizing the claim as true when presented to you, I am trying to get you to take it seriously and actively make use of it, by trying to rigorously prove things rather than produce sloppy verbal arguments that feel like a proof, which is possible to do for things that aren't true.

For all cases, if induction doesn’t fail at n=2, doesn’t mean induction doesn’t fail. Conversely, if induction fails, it doesn’t mean it fails at n=2. You have to carefully look at why and where it fails instead of defaulting to “it works at n=2, therefore it works.”

This is accurate, and related, but not the entire point. Distinguish between a proof by mathematical induction and the process of attempting to produce a proof by mathematical induction. One possible result of attempting to produce a proof is a proof. Another possible result is the identification of some difficulty in the proof that is the basis of an insight that induction isn't the right approach or, as in the colored horses examples, that the thing you are trying to prove is not actually true.

The point is that if you are properly attempting to produce a proof, which includes noticing difficulties that imply that the claim you are trying to prove is not actually true, you will either produce a valid proof or identify why your approach fails to provide a proof.

Do you think that my interlocutors were arguing this very point? Or do you think they were arguing to put me back in my place, like TheOtherDave suggests, or that there was a similar human issue that had nothing to do with the actual argument?

No, your interlocutors were not arguing this point. Their performance, as reported by you, was horribly irrational. But you should apply as much scrutiny to your own beliefs and arguments as to your interlocutors.