No, but it was you who described the person who pointed out the mistake as a pedant, which they're not, so you DO need to be corrected.
The joke does not work unless you use the word "tool". It's pointless to say "a spanner is a tool". Everyone knows that, but if you said to someone, "how's your spanner?" they wouldn't see it as funny, whereas if you said "how's your tool?", they would. The word HAS to be used, not some generic description referring back to it. A spanner could just as easily be described as an instrument, implement, or anything else.