On 19a the mishnah states if it starts raining before naitz then we dont have to complete the fast. However on 6b Abbai states if it rains kodem vatzer dtsaphrah yes bo moeshum vatser. Rashi explains "kodem vatzer"- before keriyah shemah and "yes bo moeshum vatser" hoel lo yared byom vain yapheh lolam. Since the ideal time for keriah shemah of shacharis is naitz why should rain before naitz be sufficient to stop the Tanis since as Rashi explained this rain ain yapheh lolam?
It seems to me that the same Gemara (6b) answers your question, as the Gemara concludes that only when the clouds at this time of the morning are thin is the rain not good for the world, but when the clouds are thick the rain is good. It is therefore probable that our Mishnah (19a) is only talking about rain with thick clouds, which would give them reason to call off the Tanis.
All the best,