As far as I understand the original question, you are not asking for the type of grammatical analysis that has so far been offered! It seems to me that your question is not about syntax but about verbs that have a particular purpose, right? The clue is in your guess that "to insult someone" would be categorized as a "method of conversation" and in your other guess that your group of examples would be "judging verbs".
So going on from there, I tried to think of other examples based on some of the other verbs you listed i.e. in what type of conversation would one "question someone": perhaps in an interrogation; in what type of conversation would one "compliment someone": perhaps in a review. OK, so what overall category would such "judging verbs" rightly be put in?
I did a bit of research on terms such as "assessing verbs" and "qualifying verbs" and in the end, settled on qualify. My first conclusion was that there isn't an exact accepted term in use for what you are getting at, but that it would be clearly understood if you grouped such phrases under the umbrella "qualifying verbs". Definition number 2 of "qualify" in the Cambridge Dictionary reads as follows: "specialized language. In grammar a word or phrase that qualifies another word or phrase limits its meaning and makes it less general."
Hmmmm. Then I felt dissatisfied with this answer and started looking more deeply into "verbs that evaluate", which seemed closer to what you were getting at. I found four links to offer that may be of interest to you, but the site will only let me post two so I will have to cheat the system, or you could just skip to the end if you're not interested in my process of discovery. The first link was an "action verb list for writing student outcomes" from Potsdam dot edu and I am sure you could find it if you looked it up like that. The last column is headed "evaluation" and I think that all of the verb examples you used either are in that column, or could be added to it correctly.
The next link, interestingly enough, came from the UN, and gave instructions on the correct way to reply to resolutions, in order to indicate whether you condone, oppose, or want to comment about or correct a resolution. It suggests a list of words that could begin what it calls "operative clauses". I see no reason why the verbs themselves should not therefore be called "operative verbs" and of course, there is just such a category of verbs. If you want to see this page from the UNAUSA dot org site, then search for that link plus "how to participate" plus "preambulatory and operative clauses" (phrases in quotes for better accuracy).
Emboldened by this discovery I went on to look up more information on "operative verbs" and found this interesting teacher's document from a British academy that lists a selection of operative verbs to be used when grading a paper.
So I suppose it could be said that you are talking about operative verbs that evaluate.
I became a bit frustrated by my Internet search because I found over and over again, tantalizing hints that as many as 85 classes of verbs have been identified (Beth Levin, linguist, 1993) but kept coming up against the same 4 or 5 (transitive, intransitive) on grammar and English sites, and that seemed to focus more on the syntax than the type of verb in terms of its meaning or purpose in the sentence.
But finally I came across this scholarly article about "lexical classifications" which focuses on Levin's system and does indeed group verbs together based upon some kind of shared meaning. I would say that your question refers to lexical rather than syntactical classifications.
So for example, verbs that have to do with traveling or motion would have their own class designation, while "break verbs" would all pertain to some change in the material integrity of an object or entity. This seems to be exactly the type of specificity you are looking for. I will admit I could not be bothered to read the entire 20 page paper in order to completely answer your question, fascinating though all of this is, but here is the link to the pdf. If the answer is not in there then I would suggest getting Levin's book "English Verb Classes and Alternations". Contrary to my first conclusion, there probably is, indeed, an exact term for what you are trying to pinpoint!