Yes...Now if we're talking about the OG, the fact that Cloud immediately accepts that Sephiroth is alive is part of the mystery of Cloud. If anyone should know that Sephiroth is dead, he should, but he doesn't doubt for a minute that Sephiroth killed the President or that Sephiroth is 'the real threat to the planet'. His conviction is so strong, he manages to persuade all the others. In retrospect, of course, the reason he can be so certain Sephiroth is alive is because of their connection through Jenova's cells. He knows that Sephiroth is alive and he knows he has to go after him: Cloud feels this in his bones!
In the Remake, he's haunted by Sephiroth right from the get-go, but he, and we, are left in doubt as to whether the Sephiroth he sees is a hallucination, a clone, or the real thing. He really doesn't know if Sephiroth is dead or alive or what, and tbh it doesn't matter. This time around, Aerith is the one who announces that Sephiroth is the real threat to the planet, and that they must pursue him.
And yes, Clem, many people have returned from the 'dead' in the Compilation, so naturally it would come as less of a surprise to the inhbitants of this planet. But in the OG, that norm hadn't yet been established. (Also, none of the people who returned from the dead in the Compilation were actually dead. They weren't even mostly dead.)
BTW, speaking of dead things, I was replaying the second Apps battle yesterday and I noticed that Apps doesn't dissolve into mako sparkles when he dies. Maybe he's just knocked out? I can't remember whether the other bosses you fight dissolve or not.
One of the things I LOVE about Corneo's Colosseum is that after each fight you can meet your defeated opponents in one of the rest rooms - even a very dented and battered Cutty and Sweepy.