I would say that even when speaking it is there, just that…ugh… What exactly is 'a pause' is very fluid(especially with regards to how long it lasts) and differs from speaker to speaker. ;)
At least, for me(when not reading, but just saying these), „Ona mówi” in „Ona mówi, że jestem stary” and „Ona mówi” in „Ona mówi: «jestem stary»” do not last exactly the same amount of time(still, it feels like the difference is measured in milliseconds :P).
"że" does not define anything; it just introduces whatever she might say:
Ona mówi, że jestem stary - She says I am old (she talks about me, the man)
Ona mówi, że jest stara - She says she is old (she talks about herself, she)
Ona mówi, że jest stary - She says he is old (she talks about other man, he)