During the events of Final Fantasy 7, Cloud's identity is shifted around, revealed to be based on Zack, and otherwise mangled beyond recognition. Eventually his persona is dissolved entirely and he is left as a mindless shell. In the above video, we see the moments before Tifa and Cloud fall into the life stream, where Cloud's mind is ultimately made whole again. But who is he after this?
I would say the post-lifestream trauma Cloud is the return of the old Cloud (Tifa's childhood friend), but now with the knowledge and experience of the ZackCloud. Personality wise he opens up more to his team mates. I remember this being most evident when you are on board the Highwind, and you talk to a motion sick Yuffie, Cloud's responses change. When he was ZackCloud he would dismess Yuffie's "weakness" and boasts that he never gets motion sickness. When he's back to his original self he admits to Yuffie that he's prone to motion sickness as well (as evidenced during the truck ride to Nibelheim with Zack, where he gets so sick that he removes his helmet), and even gives Yuffie some pointers on avoiding motion sickness.