Thanks to the magic of YouTube, we can do better than screenshots. This video begins during the Lifestream sequence (watch it from the beginning if you need a refresher), and Cloud's explanation about what happened to him begins around the 15:30 mark:
Cloud
"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 into SOLDIER......"
"I was so ashamed of being so weak; then I heard this story from my friend Zack..."
"And I created 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."
Barret
"Illusion, huh...?
Pretty damn strong for a 'lusion, I'd say."
Cloud
"I'm physically built like someone in SOLDIER."
"Hojo's plan to clone Sephiroth wasn't that difficult."
"It was just the same procedure they use when creating members of SOLDIER."
"You see, someone in SOLDIER isn't simply exposed to Mako energy."
"Their bodies are actually injected with Jenova cells......"
"For better or worse, only the strong can enter SOLDIER."
"It has nothing to do with the Jenova Reunion."
"But weak people......like me, get lost in the whole thing."
So the things that need support here are the following:
1) Cloud is physically like someone in SOLDIER
2) Cloud, as well as the other "Sephiroth clones," are exposed to mako energy and injected with Jenova's cells
3) Members of SOLDIER receive the same treatment described in Point 2
4) Despite the Reunion theory's experiment involving subjects who received the SOLDIER procedure, there's a fundamental difference between them
5) Weak-willed people "get lost" in the SOLDIER procedure
Point 1 is pretty much proven from the moment the game begins. Cloud is physically superior to anyone who isn't a SOLDIER, and people remark that he has the "mako eyes" that members of SOLDIER also carry.
Point 2 is proven by the following video (ignore that Cloud was named "El Boner" in this person's playthrough):
The flashback with Zack shows he and Cloud being exposed to mako. The Escapee Reports read at the back of the library after the flashback further detail that Mako Radiation Therapy was used, and that Jenova's cells were involved as well. The point is also made that Cloud had a reaction to Jenova's cells related to the experiment while Zack did not (that's important to Points 4 and 5).
The mako exposure element of Point 3 is proven by quite a lot, most obviously the "mako eyes" that SOLDIERs have, and Sephiroth's remarks from Nibelheim about SOLDIERs being exposed to mako. The Jenova cell element of Point 3 is proven by the owner of the accessory shop in Junon, who mentions being a former member of SOLDIER, and who -- as of Disc 2 onward -- feels compelled to don a black cloak like the "Sephiroth clones" despite such attire being out of season:
(go to the 3:40 mark)
Unlike the black cloaked "clones," though, he isn't forced against his will to go to the Reunion. The effect of Sephiroth's call on him goes no further than making him dress like this. That he is a former member of SOLDIER and isn't forced to the Reunion like the other black cloaks is also relevant to Points 4 and 5.
Also related to this topic, Lucrecia -- who has Jenova cells inside her body as well -- mentions that she has been having dreams of Sephiroth, and wonders if he may be alive:
(2:45 mark)
So we have Lucrecia dreaming of Sephiroth and a former member of SOLDIER feeling compelled to dress up like the experiments from Nibelheim in the same time frame that those with Jenova cells felt the call of the Reunion.
It's also noted that Cloud himself felt the call. Hojo implies that he should in the 5:25-5:50 area of this video:
Hojo also mentions a "certain hypothesis" (the Jenova Reunion Theory).
Sephiroth himself later asks Cloud if he's going to the Reunion around the 2:10 mark in this video:
Later still, Cloud acknowledges that he wasn't just chasing Sephiroth -- but actually being summoned by him:
(around the 19:20 mark)
Moving on now to Points 4 and 5, start watching that last video around the 17:00 mark for a refresher on what the Jenova Reunion Theory was all about. As you can see, it was about proving Hojo's theory that, once dismembered, Jenova would seek to reunite.
In order to test this, members of SOLDIER couldn't be used, as they do not lose their autonomy when exposed to mako and Jenova (recall that Zack did not have the desired reaction according to the Escapee Reports, while Cloud did). Though both SOLDIER and Hojo's experiment used the same procedure on the subjects involved, SOLDIER requires people of exceptionally strong mind (e.g. a self-confident, congenial guy like Zack rather than an insecure introvert like Cloud; someone okay with who they are, not desiring to be someone else) who can handle the adverse side-effects rather than losing their sense of self. For the Reunion experiment, Hojo wanted ordinary people who actually would "get lost in the whole thing" and be taken over by Jenova's Reunion instinct.
That being the case, when Hojo went to Nibelheim following Sephiroth's massacre of the village, he had the survivors (Cloud and Zack included) rounded up for use in his Reunion experiment. Recall Zangan's letter to Tifa:
(around the 0:45 mark)
Assembled, these pieces paint a very clear picture.
I can certainly understand lingering confusion from a first playthrough, but here we are 16 years later.
Not just the story, but even the game code has been dissected backward and forward -- there's not much in the way of mysteries left.