The analogy I've always used is that the closed interval has what amounts to a lid or a cap on it, while the open interval does not. Another (sort of related) way of looking at it is that the closed interval has a maximum (or minimum) value, whereas the open interval, despite only missing a single value, suddenly has this feeling of continuation, because without that final value it now asymptotes to the end and you will always be able to find a larger (or smaller) value than any previous value you examine.
I learned Math in a french-system school and yes, that's definitely a much better notation in my opinion. The difference between [] and () is not immediately clear, whereas [a,b] versus ]a,b[ makes it obvious that one includes a and b while the other does not. It also makes it easy to remember "open" and "closed" and what they mean in terms of whether or not the interval bounds are included or excluded
I learned it one way in Middle School and the other in High School. What's perplexing is that the schools were literally next door to each other, and in the same school system too.
Even though this is also the way I think of it, this underscored to me the fact that you can make up plausible explanations for anything; even for words that are opposites and are trying to explain the same thing.
