В английских интернетах общаюсь лет 7, все от друга немца, до кучки знакомых янки, до сраных индусов на ютубе - 'by' говорят. 
Я хз кто говорит 'on' в контексте конкретно 'accident'. 
Может британцы, я хз.

Только американцы и только в устной речи, согласно этому 
<a href="http://joyreactor.cc/redirect?url=https%3A%2F%2Fwww.grammar.com%2Fon_accident_vs._by_accident%23" target="_blank" rel="nofollow">https://www.grammar.com/on_accident_vs._by_accident#</a>:~:text=As%20we%20see%2C%20on%20accident,to%20indicate%20unintentionally%20or%20accidentally.

Как ежедневно спикающий с нейтивами, полагаю "случайно" и "в случае/в событии". Также как "on list" и "on Internet" здесь on это "в". 
А тут вопрос рассмотрен с научной точки зрения <a href="http://joyreactor.cc/redirect?url=http%3A%2F%2Fwww.inst.at%2Ftrans%2F16Nr%2F01_4%2Fbarratt16.htm" target="_blank" rel="nofollow">http://www.inst.at/trans/16Nr/01_4/barratt16.htm</a>

Не...я участвовать в этом не хочу.... 
Я указал на схожесть, так что всё - Финита...

Прям как проблема с "В/На Украине"...

Явное двойное самоубийство

well, a surely can say, they killed each other IN accident.

Вот что гугл мне сказал 
 
On Accident vs. By Accident 
 
A less commonly used variant of by accident, on accident is almost exclusively used in the United States. Even there, no one uses ‘on accident’ in writing, its only a spoken English term. In fact, it is considered incorrect to be used in written form. 
 
Well, most younger people tend to use on accident on similar lines of ‘on purpose’ although the meaning is completely opposite. The word ‘on’ is the confusion creator. 
 
So, we can say – “The glass slipped from my hand on accident”, however it just doesn’t seem or sound right. 
 
However, when there are other words following or in between on accident (like on _____ accident), the context of the sentence changes and it would be right to use the same. For example: 
 
There was no news report on the accident that happened yesterday. 
 
 
The head-on accident led to several injuries and two deaths. 
 
As we see, on accident cannot be used as an adverb phase, both on and accident are used as independent words in sentences. Whereas, ‘by accident’ is the only correct phrase to indicate unintentionally or accidentally.

Ничего не понял, но очень интересно

First, shoot his knee, you know.. like, by accident

On accident неправильное употребление By Accident, предположительно сформировавшегося от словосочетания on purpose и его используют в словесной форме, т.к. нету ни одного письменного источника в котором употреблялось выражение On accident. Единственное правильное употребление словосочетания On accident, когда On является частью другого словосочетания.

Our investigation shows, that he was shot on the pistol.

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