Yes I have tried the code without SLEEP as well and I get the exact same behavior. The INPUT_PULLUP indeed is not required in my case. Even when stripping the code to a minimum, changing to a new Uno I see interrupts twice. So there must be something which I am doing wrong?
Please post that stripped down code.
And just to get a clear statement: the scope pictures show the signal of the pin 3 on the UNO and not a signal from somewhere else in your circuit. Is that correct? If not, post schematics of your setup and explain where your got the signal.