Is there a term for the sort of words that sometimes precede dialogue or quotations, like say, ask, write, answer, cry, and reply?
I think they are called "dialogue tags." Have a look here: https://www.visualthesaurus.com/wordlists/9457
In Latin grammars the term verba dicendi (verbs of saying) is used in the meaning of all verbs that can be used before direct or indirect speech.
I'm having trouble finding any definitive sources that back me up here, and I am aware that Wikipedia is not an academically-acceptable source, but its page on Dialogue in Writing seems to indicate they are called identifiers (because they identify the person who is currently speaking in a dialogue).
TV Tropes identifies the use of words other than 'said' as a 'Said-Bookism', but that's more referring to the concept of using those words than a name for the words themselves. I'm afraid that's all the information I have at present.