Mozart: the French and Francophones are so
precise aren't they?

Perhaps not, or no more than we are. Suppose the word for 'to sting' (French 'piquer') referring to an insect, referred to the itch, the reaction, rather than the wound itself so 'to sting' was to cause that reaction then what we say is logical.
In truth, English speakers say bitten
or stung for the mosquito; that demonstrates a typical refinement in English which, some might say, is lacking in French (not me: I have to live there

)
Here's a nicety. Please ask Madame Mozart what words are used in French for a
snake biting

I'm prepared to hazard a guess that , whilst she will say 'mordre' ( to bite) she will admit to 'piquer' (to sting) being used too. So an animal with no stinger can sting (French) but an insect with no teeth can bite (English)

Of course, the French may say that piquer is the word for pricking with a pin or injecting with a needle so they are concentrating on the means of penetrating and or the
wound being a single punctate one.That could be reverse reasoning. Piquer for 'to sting' of an insect, could have come first, before its use for to make a needle wound,a prick with a pin, to inject, that is, it referred originally to the sensation, the itch or pain, not the wound, the injury .Thinking about it the Frenchman saying 'piquer' as in a mosquito bite, for a
snake biting is thinking the same. There is a stinging sensation or other such sensation, a reaction to the venom that we feel, when we get bitten by a snake.
We get 'stung' by a wasp or bee (so do the French who are , you could say 'piqued' by it; piquer is where 'piqued' came from , after all ) but not 'bitten' by one.This may show us as logical on another ground too. We realised that these insects have a mechanism, a sting(er) (in French 'dard' or 'aiguillon',literally also 'spear' or 'needle') evolved specifically for that purpose whereas a mosquito does not. On the other hand the female mosquito has a mechanism for piercing skin and drawing blood; that is not a stinger, evolved for the purpose of defence or for hunting ; but akin to a biting mechanism.
The horse-fly has mouth parts which are like a V-blade on a chisel and gouges a groove when it penetrates your skin. The wound from that must qualify as a bite; look at the damage, never mind the mechanism employed and its lack of dentine

. No wonder it hurts so much by comparison to the mosquito's when the wound is made. You really feel it whereas watch a mosquito bite you and feel nothing until the chemical reaction starts. It doesn't really
sting though, because it brings no irritant chemical or reaction with it to cause that sudden itch.What's more, English people say that a horse or human has been
bitten by a horsefly rather than 'stung' by one (or at least the horsey folk here around Newmarket, the centre for horseracing do). So we draw the distinction between cause and consequences.
OK ? (As the French say). Hope this subject does not cause 'le stress' (ditto) between you and Madame !