The Singularity is more analogous to the event horizon of a black hole; that used to be called the Schwartzschild singularity, but since its singular behaviour was an artifact of co-ordinate systems (as was first clearly shown by David Finkelstein), this terminology has fallen out of use.
Calling an "event horizon" a "singularity" would be pretty bonkers.
No - the commonly-cited justification is that so our model of the spacetime supposedly breaks down in the middle of black holes, so the model of the future breaks down when contemplating smarter-than-human intelligences.
Personally, I think that is nonsense. Some people are proposing busted models, is all. If your model is broken, that is hardly a cause for celebration - you should instead return to the drawing board, and build a better model.
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 ...
If you have graphic design experience, check out the on-going logo design competition at 99designs for the Singularity Institute. There are still 6 days left to enter and be eligible to win the $295 prize if your design is selected. Tell your friends with graphic design experience too. There are very few submissions currently.
Note: This is a blind contest. Designers can only see their own entries. All designs will be revealed when the contest ends.
If you're interested at getting a peek at the designs, they will be online after the competition is over. This is standard practice in 99designs contests to prevent designers from contaminating each other and having all the designs drift in a certain direction.