Another Wiper motor wiring issue - always ON no matter the key position

After skim reading through the manuals and online forums everything is pretty simple. Now i did change the wiper motor with another spare one i had and it works!!!

I'll try to post clear photos of me dismantling the broken wiper motor and repair it (hopefully). For reference in case someone has this problem again

Now my new question is : do I need to connect the "C" Terminal wire from the bulkhead to the neutral safety switch wire? like in the manuals

The reason i ask is because the wiper motor still functions perfect without it. I could simply ignore it and call it a day BUT i hate not figuring stuff out.

there is a saying a lot of old timers quote : "there is always a reason behind everything"