If you do not match or connect the wires based on position, when you plug the relay into the connector on the new pigtail, it will not be connected to the correct wire. The spades from the relay are positioned such that it is impossible to insert them into the connector in the wrong position. For a reason.
Looking at the last picture in your series, the one connector with a relay or something plugged in has had ALL the wires cut & spliced and wrapped in electricians tape. If those connections are not soldered, they might be the cause of your problem. Another wire near there looks as if it has been connected using a crimp style butt connector. Is that connection good also? I'd be checking all these 'repairs' before replacing stuff. The 'corrosion' you mention does not look to me as if it would prevent anything from working.
IOW you are making a repair that is not needed, IMO.
tom