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In other words, it simply means, forks happen due to opposing beliefs about a certain proposal to improve the system (or other reasons) which, as a side effect, could weaken the "original" (or whatever you may call it) system. That's hardfork. Softfork, on the other hand, is about adding new rules which needs approval of the majoriy of the miners.
Like a game. Adding new rules is called softfork while creating a new game inspired by the original one but works differently if compared to is called hardfork.
all wrong. you are explaining a split but using the word "fork".
a fork is simply an upgrade to the code. some addition or deletion of rules for example. that is all. the split happens if the fork doesn't have enough support. and it doesn't happen if it has the support of the majority.
your definition of the soft and hard fork are also wrong. although there is not a clear definition that everyone agrees, but generally a soft fork is when the software remains backward compatible and a hard fork is when it doesn't. both can add new rules.
Still, what are the implications in terms of volatily?
volatility and price are speculation related. they have nothing to do with the forks. they are more like the reaction of the market to some news.
If I understand well, SegWit helped Bitcoin to improve so the coin raised its value (able to deal faster with a higher number of transaction)
SegWit didn't "improve value" the market did. SegWit improved bitcoin and added new functionality and that plus the end of scaling debate led to price rise.
while BitcoinCash would like to crush Bitcoin (whose value would plummet)?
that is politically correct not technically