Hi there

We have a door lock, exterior door, that I need to replace.

There is one deadbolt, with a keyhole on the outside and a knob on the inside to lock/unlock... this one is unrelated to the one that needs replaced.

The one that needs replaced has a keyhole on the outside, and a horizontal handle on the inside. If the horizontal handle is moved down, it unlocks that lock. It does not interact with the deadbolt described above.

The handle-lock also allows you to insert a key on the outside, and turn it to the left, with the internal handle DOWN, and remove the key - leaving that latch in the 'unlocked' position where anyone can push/pull the door open (if the bottom deadbolt is not locked).

I've tried searching for paddle lock, handle deadbolt, and everything else I can think of but can't find one of these... do they have a certain name? Where can I buy a replacement?

Most likely just called an entry lock. Various designs by various manufactures. Look in front door entry locks or just entry locks.