Oddly, both grammar point have a “だす vs はじめる” link under reading but it’s two different resources. Either way there are enough examples there if you want to read about the difference in nuance.
The short story is that だす is more sudden, focused on the start of the action, and somewhat uncontrollable (that’s why it’s often used with emotional actions).
Maybe the hint could say that, or maybe the English version could say “he” instead of “I” to make だす seem more likely…
So let say we want to say somebody “started to grow up”.
始める would mean he started in general (process is important).
出す he started at specific time or event even if it not rapid development (starting point is important)?
Do i get it right?
Right, so since that nuance exists, はじめる is incorrect, and it should be marked as such.
Nuance is small, not everybody is here to learn how to write poetry 
. I would press ops straightaway. 
I you believe it is important mistake then just enter whatever.
I don’t think はじめる is wrong; rather, isn’t だす almost a bit odd because the English translation portrays it as a controllable action of the narrator?
For what it is worth my dictionary gives it a meaning of 【開始する】, and then:
開始する - begin*, start ; open. （⇨始める）
▸ 仕事を開始するbegin [start] to work / begin [start] working.
▸ 交渉を開始するbegin [start, open, commence] negotiations ｟with｠.
▸ 銀行と取引を開始する｟口座を開く｠ open an account with a bank.
So it is looking more like a starting point is important.
EDIT:
I am not skillful enough to understand it properly on myself so I will just leave it here:
～(し)始める
▸ 雨が降り始めた
It began [started] to rain. / It began [started] raining. (!begin, start とも不定詞・動名詞の両方が可能だが, 不定詞は行為・運動の開始自体を, 動名詞は開始後に引き続く行為の継続的遂行を強調する)
▸ 彼女は会議のあと疲れを感じ始めていた
She was beginning to feel tired after the meeting.
▸ 先生が話し始めると皆おしゃべりをやめた
The teacher started to speak, and everyone stopped talking.
Both use “start to rain” as example both here is somewhat complicated explanation. I will not even try to start to decoded it xD
I don’t think there’s any reason it has to be uncontrollable. If you look at words like 走り出す, 言い出す or 考え出す, they’re all controllable actions.
There is 話し出す vs 話し始める as well. An many others.
I would guest fist one is describing exact moment somebody open his mouth and start talking (somebody eg suddenly open his mouth during silence and scared you) and second put stress on the fact it started talking (we don’t know for how long and we don’t care that much about moment of beginning but we care about the fact this action started and continues).
That is all I could guest by analysing entries and examples in dictionary. Explanation given by dictionary seems to say something like that, but I am not able to fully understand it ((し)始める !begin, start とも不定詞・動名詞の両方が可能だが, 不定詞は行為・運動の開始自体を, 動名詞は開始後に引き続く行為の継続的遂行を強調する)).
That would explain problem with translation. “suddenly” looks like simplest way to put stress at the fact that beginning of the action is the most important. In this sentence fact that he started before “cheers” seems to be important. Fact that he is drinking is not.
But there is 泣き始める.
I am even more confused now…
But It could kind of make sense (i guess?) in framework I tried to put together:
If you said something and somebody started crying that would be sudden so: 泣き出す
But if somebodies partner died and that person is sad, and started crying seeing that person picture then 泣き始める since the fact it started is not important nor surprising. Fact that that person is sad and crying is more important.(or it is other way around? I don’t know anymore…)
Anyway: I burnt too many brain cell on that problem…
@Asher @mrnoone Senseitachi-dono! Do you have any input on that?
出す originally means “to produce something (out of nothing)” “to release”. By extension of that, it has come to mean “to start” something, but with a nuance of something happening abruptly and regardless of the speaker’s will.
始める doesn’t have those nuances. It simply means “to start” something.
I think that a widow suddenly bursting with tears at a funeral is good example expressing both abruptness and uncontrollably. 出す is also often emphasized with expressions like どっと or 突然 (suddenly).
Because generally verb[stem] + 出す verb is “beyond” will of humans, that is cannot be controlled, it usually feels unnatural to ask someone to do verb[stem] + 出す action, while it is ok to ask the same with 始める, in other words the 始める action can be controlled. Also, commands don’t work well with 出す. It’s hard to ask someone to laugh, and even harder to order him to do so (in normal circumstances 
). Because of this nuance 笑い出す is much more common than 笑い始める. Laugh is (usually) beyond our will and abrupt.
Example:
話し出してください Please, start speaking. <— natural
話し出してください <— less natural
Of course, the choice in the end (between 出す and 始める) belongs to the way how the speaker sees the situation.
I hope it helps,
Cheers
xD Less natural one would be 話し出してください？
Thank for time and effort. I will probably start collecting sentences with those forms to force my brain to grasp the difference in more organic way. But your explanation is good start. Thank you.
I used to feel the same way, frustrated at the multiple seemingly correct answers. However, as I advanced in my studies and started to learn the subtle differences in nuance, I began to appreciate more and more the thoughtfulness that was put into building the sentences.
I find myself thinking “A could work here, but B better conveys the feeling that…”
So I grew to like the way the system doesn’t mark it as incorrect but moreso needs improvement.
I think there’s a typo here…
話し始めてください <— Please, start speaking. <— natural
話し出してください <— less natural
There’s a difference between being controllable by the person doing it, and being controllable by the person talking about it. That’s why I wrote that it would be more obviously 出す if the English translation made the sentence about a different person than the narrator.
@FredKore @MacFinch @nekoyama
I have meant:
話し始めてください Please, start speaking. <— natural
話し出してください <— less natural
To answer @ggw1776’s original question, the reason that sentences prompt a retry instead of counting those answers as “correct” is so that you, the user, can’t potentially add a grammar point, then successfully “pass” all of its reviews without ever actually using the grammar point you’re supposed to be practicing.
In this example,
If I answered all 12 of 出す’s reviews with 始める instead, getting a “perfect” score and clearing it entirely from my reviews… I don’t think I would’ve actually, properly “learned” 出す at all.
So, BunPro lets you know when what you’ve tried is valid (albeit maybe with a different nuance/meaning), but is trying to get you to practice something else.
After some thought I think I wasn’t that much off the mark. 
My assumption that 始める would need to have some contrasting nuance got me into woods.
If I understand you correctly: by 話し出す you mean something like “spoke out” (somebody just said something and the fact it happen - it starting point - is important because it sudden or unexpected) or something like “start saying something without thinking or control” (that person didn’t control what they say) or any combination of both.
話し始める you just meant they start talking and it continues. No nuance added.
That would explain everything to me actually. 泣き始める would be just very plain or (because 泣き出す is so common) it could even carry the nuance that we are not sure if that person is not pretending (actor on the stage would rather 泣き始める than 泣き出す if we are not immersed into the story).
It even make sense with weather: 雨が降り始める is fine and plain (we didn’t pay attention that it just started so no need for nuance or we even believe 神様 ordered it rain and it fallow that order - animalistic approach), but 雨が降り出す would add that extra layer of surprise, or suddenness, or uncontrollability.
Am I close? 
Edit:
My example with growing up would work too. Using 出す would imply something sudden started the process even if the process is slow. Eg my father died and I had to provided for my little sister. So it was sudden in a sense and outside of my will. It seems to make sense.
Right, and I guess my original point is I would rather just have it marked as incorrect
I believe, if I understand it correctly, that he just said both are grammatically correct and the minute nuance depends on speakers believes. Two Japanese people could say the same sentence in different way in the same situation. It is smalls semantical/cultural difference. Not grammatical.
Additional clue would be nice though .
Maybe another option in setting would solve your problem?