"Pissen" is vulgar language and not to be used anywhere. "Pinkeln" is a colloquial word that might be used within a family, mostly with children. Kids will also use it among themselves. When talking to the doctor, you might say "Wasser lassen". And the doctor will probably write "urinieren" in his report.

Both verbs are very informal. I'd say pissen is a bit more rude. A very common term which is often used between doctor and patient is "Wasser lassen".

Both words are extremely informal and never used in medical issues!! I'd recommend to say ,,auf die Toilette gehen" (going to the toilet) and at the doctors you would probably hear ,,urinieren" (to urinate).