They start showing up after you kill the dragon.
I believe you're misinterpreting what Elder Scrolls Wiki is saying.
The first part of this quote makes it clear that it is talking about completing the quest. It says you should avoid completing it because the dragons can be a hindrance:
Waiting to complete this quest until later in the game can be beneficial. The random dragon attacks can kill non-essential NPCs who still have activities or quests to be completed, make travel difficult, or simply become an annoyance with the frequency of attacks.
I can see how the latter part of the quote could be confusing:
In order to avoid this, the previous quest Bleak Falls Barrow must not be completed until the player is ready, as Dragon Rising starts automatically upon presenting the Dragonstone.
But, note that it is not saying that the attacks start when you present the Dragonstone, but that Dragon Rising starts. This is important, because the beginning of Dragon Rising starts a series of inconvenient quest triggers that eventually lead to the end of Dragon Rising.
Bringing the Dragonstone to Dragonreach starts the conversation between Delphine and Farengar, which leads to Irileth telling Farengar that the dragon has attacked, which leads to the dragon showing up at the Western Watchtower.
I believe that the city of Whiterun is pretty useless due to panic during Dragon Rising as well, but would have to double-check that.
So, since you don't want to avoid ever entering Dragonreach, and after that you don't want to avoid the Western Watchtower for fear of triggering a dragon, it's best to just not even start the quest, lest you be forced to finish it.
As further confirmation, note that that same wiki page explicitly says:
After completion of this quest, Dragons will begin appearing in the wild and attacking the Dragonborn at random.