I didn't give birth until one day before EDD. It's still considered acceptable to wait till 41-42 weeks but it also depends on your gynae's recommendation.
I was actually due for a follow up and was considering a membrane sweep on w39+5. My water broke at 1am on the day of my appointment so I was admitted instead.
How I differentiated pee from amniotic fluid was by smell. Amniotic fluid should be odorless (mine was) or sweet smelling (according to the internet haha). Pee should have a distinctive amonia smell to it. Also, if it's from coughing it may well be pee actually. Another way to tell if it's amniotic fluid is whether it leaks out when you have positional changes, such as sit to stand or lying down to stand. Leaking amniotic fluid usually pools in the vaginal and leaks out when you move, whereas pee usually just leaks out. I tried to test by changing positions too and my fluid leaked everytime I stood up. I absolutely had no contractions until I was admitted. My dilation was at 0cm when I was being screened at the delivery suite.
Hope all goes well and you have a smooth delivery!