The event horizon is a co-ordinate singularity--like the North Pole. (Wikipedia has an article on the types of 'mathematical' singularity. I haven't read it, so I don't know how good it is.) Yes, in some sense it was a mistake to call the event horizon a 'singularity'; that's why they stopped.

Otherwise, we appear to agree. As you say, 'If you divide by zero in your model, that just shows that your model is broken.' I share your interest in good terminology; but it looks to me like it's probably far too late to get rid of 'The Singularity'. So I just adopt it, but with a certain air of amused detachment. Sure, it's just a co-ordinate singularity at most.

I wonder if Ulam was thinking of the Schwarzschild singularity in 1958; that's the same year Finkelstein's co-ordinates were introduced, so he may or may not have absorbed the illusory nature of the horizon at that point, but probably he wasn't thinking of the singularity at the centre of the hole. But I'm not much of an historian, and would welcome corrections from those better informed.