You’re missing a few things.
Other way around. Think about it. If people save more, that means there’s more money available to be lent out…that is, the supply has increased…thus meaning the price (i.e. interest rate) would go down.
How did you get to that?
If you pay a bank a fee for storing something, it’s just that…a fee. That’s not your money anymore. That’s their money. They earned it. Would you consider it “saving” $100 every month because that’s what you pay Public Storage every 30 days for providing you a unit to keep your boat in?
Here’s a couple of articles that address “saving” that might help:
Also, for more info on interest rates, check out this recent thread. There’s plenty in there that will probably help.
P.S.
Welcome to the forum! Be sure to beginner’s thread.