It's pronounced like "cock". It's named after the animal, and that's how you say it in French. It's how people who use the tool say it.

Not really, the English word "cock" has a very different vowel than the French "coq" - https://youtu.be/bIraCzL3dtI?t=4s is a pretty fair exemplar. It's a lot more like the "o" in "joke".

(edit: maybe you're English? Some English accents of England would articulate "cock" very close to French "coq" and "joke" would be far away. Apologies, one really does need to specify which "English" is meant, and I should have seen that at once. First cup of coffee still in progress.)

Quick look on YouTube, this guy says "cock" [short staccato 'k' at the end] https://www.youtube.com/watch?v=WFen8x66ETY and this guy https://www.youtube.com/watch?v=sW4uki0Fr8I - both apparently French (though the guy on the second one really sounds like he's faking it!).

In your example the 'o' is more precise and the 'k' is swallowed a little.

I'd expect they're different accents, perhaps your example is more Northern French (I really don't know).

Perhaps more like "Coke"?

I pronounce it as "coke", for fairly obvious reasons :-).

So there is this rumor hanging around the french community that use Coq, that the name was deliberately chosen to sound like "cock" in english. This way, french researcher could brag to other collegues about going to a conference about "cock", in all seriousness.

Something like this: https://www.youtube.com/watch?v=XKUE62hOvHM