Serious mode for a moment: transpire typically means "to happen" or "to occur". The "to shed moisture or vapor" definition is now usually limited to discussions about plants (moisture) or the soul when speaking of death in a figurative manner.
Perspire means "to shed matter" but is most commonly used in reference to sweating. The reason that the term referring to "matter" as opposed to the one referring to moisture is preferred is likely due to the salt content in sweat.
Both words share the Latin base for "to breathe", spirare, which is also where we get expire, aspire, conspire, and even spirit.
Back to the joke: you were correct with the way you used transpire in your reply. I "miscorrected" it to make a pun that tied it back to sweat.
Yup! Spirare is the Latin root of a lot of words, most likely due to how pneuma was believed to be the human soul. Breath and soul were one and the same, so it shows up in a lot of places.
I've never heard of transpire being used for the first definition. Perspiration is the act of sweating, and it comes from Latin I think? High school was so long ago.
Thanks!
I think @lemonmelon explained it well. Seems I was skirting the definition by a whisker and got away with it but still failed as a joke or pun should be understandable without having to pull out a Latin dictionary 😀