I never realized that the British definition of irony included acting in a manner contrary to one's stated preferences.
I have heard the "paradoxically ironic" bit from other people before and it never made any sense to me. A lack of irony is not the same thing as the opposite of irony. I have also never been able to understand the difference between "ironic" and "paradoxically ironic."
See, for example, http://www.buzzfeed.com/ailbhemalone/the-19-most-ironic-fact...
Granted, it's not the most scintillating use of irony - but it fits the British definition quite adequately.
Of course, the ironic thing about Alanis Morrisette's song is that it contains no irony. Which makes it paradoxically ironic.