Timeline:
-Cloud leaves town to join SOLDIER.
-Cloud returns as a grunt, with Zack and Sephiroth. The scenes from Cloud's past (the visit to Nibelheim) are all true, except Cloud puts himself in Zack's place. He says HE is the one in SOLDIER, but it isnt true. Cloud was there, but as a grunt. HE is the one who fights Sephiroth and saves Tifa, and is then taken to the basement with Zack and experimented on.
-Zack and Cloud escape to Midgar.
-Cloud meets Tifa in Midgar and the game begins.

YES all of his childhood memories are true, save for the fact that Tifa and Cloud were never very close. They grew up together, he liked her, he wanted to impress her and so on. If he didnt know Tifa, why would she know him?