Question. I have read 2 different version of how the city was captured

One was that a gate had been left open (either by accident or by a traitor) and the Ottomans entered the city that way.

The other was that they breached the wall with cannon and overwhelmed the defenders that way.

Warren Treadgold in A History of The Byzantine State and Society mentions both events happening. My impression from that and reading John Norwich's A History of Venice was that both events were near simultaneous. It's my understanding that what happened was as follows:

1) Giovanni Giustiniani Longo commanded the overall defense of the land walls. He held the breach in the outer wall against several assaults but was wounded by a crossbow bolt.
2) He loses heart and wants to withdraw.
3) The Emperor pleads with him to stay but Longo won't.
4) The Genoese carry him away. They make up the largest contingent of Italians defending the city. Those Genoese still fighting think all is lost and withdraw too. Severely weakening the defense of the breach.
5) This leaves only the Greeks and Venetians defending the area around the Palace of Blachernae.
6) Turks appear on the walls. As you said the Kerkoporta had been left open.
7) With Turks on the walls, the rest renew the pressure on the breach, forcing their way through.
8) Emperor Constantine sees all is lost and dies in the melee.
9) Venetians rout/withdrawal towards the docks and many are taken prisoner including the Bailo, Girolamo Minotto, whom is immediately executed on the spot.
10) Everyone flees as resistance ends.