Crash is right, those examples are not irony. Contradiction is not irony.
Alanis was also incorrect- but oddly enough, in a way that works. Because the fact that her song was called 'Irony' and entirely failed to contain a single piece of irony is a little bit ironic!
As Greg Proops says, rain on your wedding day is not ironic, it's a bummer. It is only ironic if you were a weatherman who predicted good weather, and even then it wouldn't be THAT ironic.
Alanis herself happily admits her error, though, and points out she is a singer, not a grammatacist, and if that is the worst thing you can find about her music she is pretty happy.
Despite the fact that irony does ALSO mean 'like iron'- though no-one uses it that way- our pals at Oxford- pretty much th centre of the English language- describe it thusly:
n. Expression of one's meaning by language of opposite or different tendency, esp. simulated adoption of another's point of view for purpose of ridicule; ill-timed or perverse arrival of event or circumstance in itself desirable, due to the feigned good will and actual malice of (Fate, circumstance etc.); use of language that has an inner meaning for a privileged audience and an outer meaning for the persons addressed or concerned (sometime sincluding the speaker)
That last bit is what is happening if you say that someone is speaking with deep irony (e.g German Jew who runs a bank saying how good Hitler's War is for the economy- delivered straight that would be inanity, delivered ironically it is cynical wit- but it only works for those who know all the facts). So that is distinctive from a situation being ironic, which is what people here are trying to describe. The mistake people are making is that because irony normally involves contradiction, therefore contradiction is irony. This is a common problem with grammar...