French native people are aware of the different meanings. Here, the context makes clear coached is intended.

Édouard laugh "like a girl" and his sister train him to do it a more masculine way.

_{This answer comes as a complement to the concise and correct one provided by @jlliagre.}

1. For a native French speaker, is there any indication which English term would be more appropriate?

In certain cases, such as this one, a French word can be translated into several different words in English. What allows us to differentiate them and chose the most relevant one is the context.

With the context provided here, I would actually use the verb to train, because it looks like Edouard’s sister is trying to teach him how to shape his laugh so that it sounds less girly. Therefore, the best of your two suggestions is to coach in my opinion.

2. Should I conclude that entraîner simply has a wider meaning than the English terms and that a French speaker will sometimes (perhaps often) be content not to define a situation as precisely as an English speaker would be required to do?

Entraîner doesn’t have a wider meaning, it has several different ones. The phenomenon of a word having several meanings, and therefore multiple possible translations in another language is very common and not specific to French. I even looked for a technical term to describe it but it looks like they couldn’t find one on Linguistics SE.

Examples also exist in English (to get, to make, etc.).

As I said earlier, it’s all about context, and whatever the language, this is what allows people to understand what meaning is the correct one. Situations where the context is not clear do exist, and they lead to the concept of ambiguity.

I First question

The meaning of "entrainer" in this sentence is neither "encourage" nor "coach".

From the TLFi (definitions have been changed to bold type)

I.− [L’idée dominante est celle de mouvement imprimé à qqc. ou à qqn]
C.− Au fig.
c) [Le suj. désigne une chose abstr.] Emporter, pousser quelqu’un vers quelqu’un ou vers quelque chose sous l’effet d’une influence irrésistible
2. [Le suj. désigne une pers.] Amener quelqu’un à agir, à faire quelque chose en exerçant sur lui une contrainte, une pression morale
► Comment le supposez-vous en même temps assez amoureux de la guerre pour y entraîner ses chefs, qui ne la veulent pas? (Renan, Drames philos.,Prêtre Némi, 1885, III, 1, p. 569).
► Il sait comment on entraîne les peuples et comment on les fanatise (Coppée, Bonne souffr.,1898, p. 105)
► Il n’y a qu’un mot à dire à ces impuissants de la liquidation : « ce qui est fait est fait », et les entraîner de force à de nouveaux destins. Mounier, Traité du caractère,1946, p. 453.

The essential of the definition, "Amener quelqu’un à agir, à faire quelque chose en exerçant sur lui une contrainte, une pression morale", means "to bring someone to act or to do something, this being achieved by exerting on them a constraint or moral pressure". Moral pressure is not relevant here. The constraint might have consisted in encouragements (not much of a constraint but still not negligible), however it is not likely that you can impart the courage to laugh, that is exhort someone to laugh in explaining that there is nothing to be afraid of. The constraint was possibly of a different nature: it could have consisted in the stimulation of their ability to emulate, which I think more probable. Nonetheless, this is not told and all that is said is that she brought him to laugh. In French this would be synonymously put as (from the definition) "…combien de fois elle l’avait amené à rire…".

ref. Memoirs of Louis XIV and His Court and of the Regency … I had brought him to laugh at his own weakness, and even to take jokes without caring for them.

The sense "II. B" in the TLFi, which would be "to coach", does not seem to apply.

B.− P. ext. [Le suj. désigne une pers.] Former, soumettre quelqu’un à une activité physique ou intellectuelle de manière à créer une aptitude ou une habitude.

It implies a rather methodic activity ("former", "formation") which is not likely here.

II Second question

As seen in the first part, "entrainer" has a wider meaning and a perusal of the entry in the TLFi should make you aware of its scope. The particular sense I’d retain in this sentence ("Édouard avait quand…") has been mentioned in "I". The second part of this second question can’t be answered in a scholarly way before a lengthy comparative study of the two languages; an approximate answer from the mouth of a professional translator could possibly have some value. Personally, I can’t confirm or infirm this contention, I do not have enough experience. What I can assert is that expression, in any language I presume, but certainly in French and in English, does not consist in formulating in all circumstances the most detailed statement applying to a situation. For instance, in English, you can talk of such and such a number of fighters put out of action, for instance, without saying precisely what is the number of the dead and the wounded; nothing keeps you then to become more precise and produce those numbers; you can as well start with those numbers and forget about the general statement. This is similar to this question of "entrainer quelqu’un à rire".

Édouard avait quand même l’air d’une fille, combien de fois elle
l’avait entraîné à rire « normalement », des séances de travail qui
finissaient dans les larmes […] (Au revoir la-haut, Pierre Lemaitre)

There are indications of a recurring event (combien de fois; des séances de travail) and this is more compatible in my view with the idea of teaching someone to do something (discussed elsewhere) using repetition, than with having something finally result in something else, or being related to someone inducing or influencing someone to, or dragging someone (along) with figuratively, or whatever. If you divide the possible meanings into motion related and training related, it’s pretty clear to me it’s about the latter; the tears are the actual consequence of those sessions i.e. the former meaning. You have polysemy but in context I don’t find the verb encourager a sensible choice for a synonym as context (many instances of an event) combined with clues (combien de fois instead of some adverb like finalement showcasing a final outcome) and finally the reference to séances de travail defining all those moments, precludes such a construction I would think. So I don’t find the sentence vague or ambiguous in any way.

Translation to English is a skill and is not your typical train of thought when speaking French so the selection of a word in English depends on proficiency/training and research. Providing an overview of the different meanings of the verb (s’)entraîner in French along with context and asking native speakers of the English language to provide a translation in English should yield better results than asking native French speakers.

The second point is moot but whether a "French speaker [would] sometimes (perhaps often) be content not to define a situation as precisely as an English speaker would be required to do" is seemingly about bias, be it confirmation or cultural; a technical presentation like this one (compare with the English language version), should help tone it down.