Translation:I forgot my umbrella at school.
i get that "forgot" and "left" accomplish the same meaning in english in the context of this sentence, but they are still two very different verbs, and the translation should be left as "forgot", not "left". you would never use わすれました for anything you intentionally leave
It may not be grammaticallly correct to use "forget" instead of "leave" here. According to Oxford Advanced Learner's Dictionary:
You cannot use forget if you want to mention the place where you have left something: I've left my book at home. [But not] I've forgotten my book at home.
Though it may be technically wrong, it's something that pretty much every english speaker would say. Even though it may be wrong, it would be nitpicking to mark it wrong, especially since it's a much better translation than "left at school", which is the important part. And I honestly have no idea how you would say "I forgot my book at home" with the same meaning, but without the use of "forget". Though yeah, you could say "I forgot to take my book from home", not sure you'd hear that as much as the former.
Just for information, as an Australian English speaker, I would never say this. To me this sounds like something really little kids say, like it's not quite right. I would always use left if I was saying the place. I have to use this a lot unfortunately, I'm always forgetting stuff
I think に indicates position and で indicates the place where something is happening. So movements happening on that position. The abrella is just lying around there, so probably therefore it's に. So if you would use both sentences, がっこうでかさをわすれてしまいました and がっこうにかさをわすれてしまいました, then the 1st would indicate the forgettingpart, so you were at school busy with forgetting your umbrella (little weird to imagine right?) and the second one would indicate more that the umbrella is somewhere in school (position), cause you forgot it.
I recommend https://japanese.stackexchange.com/questions/60/particles-に-vs-で It's hard to explain, but に is kind of like "into." I think 学校で傘を忘れる means closer to you forgot about its existence (maybe)
From what I understand: に is used with verbs of movement (e.g. 行きます、来ます) to state the destination, and with verbs of existence (e.g. あります、います) to state where something exists. で is used to state where (or when) an action takes place, or the means of how an action occurs (like what you use to do something). を is used with verbs of movement to state where you go through (e.g. a hallway) in order to reach some destination, and of course to mark objects in a sentence.