I don't know about the CC story either, but I've always gotten the impression that it's a little of both. Cloud says in FFVII that he heard about what happened with Sephiroth in Nibelheim from Zack, and that he adopted it for his own:
"I never was in SOLDIER."
"I made up the stories about what happened to me five years ago, about being in SOLDIER."
"I left my village looking for glory, but never made it in to SOLDIER......"
"I was so ashamed of being so weak; then I heard this story from my friend Zack..."
"And I greated an illusion of myself made up of what I had seen in my life....."
"And I continued to play the charade as if it were true."
After awhile though, he comes to believe his own lies, thanks in part to the experiments. I should point out that this is one of the more vague parts of FFVII, no thanks to the translation, and still comes up for debate from time to time. However, general consensus seems to be that the experiments had a strong effect on his memory.
As for Tifa and Aeris... Tifa DID recognize that something was off with Cloud's story. In FFVII, during an internal monologue of hers, she thinks:
"You told me a lot about what happened after you left Nibelheim..."
"But..."
"...Something's wrong. I felt there was something strange about the things you talked about."
"All the things you didn't know that you should. And other things you shouldn't know that you did..."
(she's clearly talking about the Nibelheim incident in that second bit; at the time of this monologue, she didn't know that Cloud was present for that)
"I wanted to make sure..."
"But then I heard... you were going far away..."
"And I didn't want that..."
"I didn't know what to do. So, I thought I needed more time."
"And that's why I told you about the AVALANCHE job. I wanted to be with you, watch you."
So basically, Tifa's in "observation" mode for a lot of the game, for better or worse, up until the point that she and Cloud fall into the Lifestream in Mideel, and she starts guiding him through his "real" memories.
As for Aeris, taking FFVII's canon alone, it's clear that she never heard about what happened in Nibelheim from Zack (or other sources) before Cloud told his story. Still, it's Cloud's similarities to Zack that intrigue her, so she's in a sort of "observation mode" herself.
(And as for my FFVII historian qualifications, I've been on the staff of a well-known FFVII fansite for several years, including a stint as webmaster. Oh, and I've played the game a bunch, too

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