Why isn't the title of this trope a pun? Hurricane of Puns doesn't fit in with other pun-related tropes like Just For Pun and Pungeon Master, so why not call the Hurricane of Puns "Punderstorm," instead?

I'm noticing in some examples the person who gave the hurricane will simply say something that fits the topic but won't work it into a sentence, such as the Family Guy example. Is this notable enough to mention in the description?