The cockatrice owes much of its currency in English to translation. Early modern English Bibles, including the King James version, use the word to render Hebrew terms for a venomous snake at several points in Isaiah, Jeremiah and Proverbs, most memorably in the passage where a weaned child puts its hand on the cockatrice den. Later English versions replace it with adder or viper, because the Hebrew names a snake and nothing more.
The effect of those two centuries of use was to give a bestiary creature scriptural standing. A reader who met the word in church and again in a herbal or a book of wonders had no reason to think the two were different things, and the cockatrice acquired the authority of the text it had been inserted into. That is one of the clearer examples of a translation choice creating a creature rather than describing one.
The heraldic cockatrice is a separate track and it is precise where the folklore is not. Blazon gives it a cock's head, wattles and comb, a dragon's wings and body, and a barbed serpent tail, and heralds distinguish it from the basilisk, which in strict usage has a dragon's head at the end of its tail as well. Outside the college of arms nobody has ever kept that distinction, and the two words remain interchangeable in ordinary use.