My definition of savings is money which is part of a steady income which is not immediately spent but rather kept aside for a specific future purpose.
If the money being kept does not have a specific purpose, then I would call it "emergency funds" or "perpetual savings", which would be meant to cover unexpected events, like medical bills, reparations or replacement of important household goods.
If the money is being kept in the form of a investment and not an stable asset, then it would serve to cover expected expenses/purchases, since the emergencies can coincide with a bear market, which would imply to sell at low price and no one wants to do that.
That's a good concept because when it comes to savings, there are definitely going to be a lot of differences in how people express themselves in this regard but that doesn't mean the goal is different because it's clearly the same goal,
I also do something like this because saving doesn't mean that we save most of the money that we have but of course only the allowance of the leftover money that we don't use for emergencies like you said.
With conditions like this, at least we can still cover when there are things that are not wanted to happen later considering we will never know the worst possibilities for ourselves in the future but with the savings intended for that, at least we can still be a little less worried later because there are funds that are already intended for that.
