Apparently, your pumps were manufactured in some alternative parallel universe of some kind, where the process of inflating a tube relies on the valve in the tube itself. In our universe we don't even need a core in order to inflate a tube: it will inflate perfectly fine, provided the built-in one-way valve at the base of the pump is working properly.

It is unbelievably bizarre that your pump somehow managed to draw air from the tube on the backstroke. All pumps have a built-in valve intended to prevent exactly that.