What is a reasonable definition for "is"? What is the rule for inserting it in a sentence?

The honest answer — especially for those who have little to no understanding of esoteric grammar rules, but also for those who in fact do — is "there is no reason". It just so happens that English is made that way.

"That not exactly correct" is a perfectly grammatical construction in Russian (or East Slavic languages in general). There, a copula is usually optional, in many cases not an option at all, and in fact for all intents and purposes you could say the word "is" as such does not even exist. (I am exaggerating just a little, and a native speaker or a linguist will understand what I mean. For everyone else, some examples follow shortly, so bear with me.)

Now, there is no intrinsic reason for English to be this way and for Russian to be the other. The only reason is "because reasons". Of course that's a disappointing answer to provide, so you actually can try to drive that point home by pointing out that there are also other differences, in the very same sentence, to which your tutor apparently is not objecting. So it's a small step for him to also not object to this one and just accept it how it is.

For example, note that in Russian (I will just go ahead and use that one as an example), the word "correct" in the sentence at hand would have to agree in gender and number with the subject ("that"), while English does not even have such a thing as inflection of adjectives for gender or number.

• "She is correct, it is correct, we are correct" in Russian would be "she correct[female ending], it correct[neuter ending], we correct[plural ending]".
• "There is a car in the street" would be "On street car".
• "It is cold outside" would be "Outside cold[neuter ending]".
• "Today is my birthday" would be "At mine today birthday".
• "There's something strange about this language" would be "With this language something strange" — where both language and this would be inflected for case, and for a case not just English, but indeed most languages have never heard of: Instrumentalis.

Note how all these examples must have an "is" in English, and must not have an "is" in Russian. From there, it's a tiny step to accept the "that not correct" vs. "that is not correct" discrepancy, as it is indeed the norm and not an exception.

And the reason, again, is pure chance. There is no universal rule that every language must (not) use copulas, or genders, or cases, or aspects, or articles, or the dual number, or a future tense. Some do, others don't. And it took them millennia to develop that way, one tiny step at a time. You can't pin it down to one particular thing.

In that sentence is is a copula.

It is used in your example to join a subject (that) to its description (not exactly correct).

I, myself, am a speaker of three Austronesian languages (to which Malaysian belongs), and I can attest to the fact that some Austronesian languages simply don't have the be-verb. For instance, "She's beautiful" in Waray--a Philippine language--would be "Mahusay hiya" (Beautiful she.)

I understand you're not keen on overwhelming your student with a lot of explanation on grammar, and you mustn't. So, as a teacher of English myself, my advice is to use the lexical approach. Teach vocabulary in chunk/ phrases rather than by focusing on individual words. In sentences like 'She's beautiful," students must accept that that small word--the be verb--simply has to be there.

It sounds like you haven't covered the basic types of verbs in English - the pure copula is one of them. It joins something you're talking about to one of its attributes, for instance, 'the stone is white'. The verb 'to be' is also used as part of tense formation, eg the present continuous, 'the student is learning'. It is used as an intransitive verb, where an action is described which starts and ends in the subject, eg 'I think, therefore I am', where the verb denotes existence. In your sentence, the verb 'to be' is used as a pure copula, with the sense 'That thing is a thing which is not exactly correct'.