It’s really up to the player. Barret is the big ‘emoter’ in the aftermath scene (and in the scenes wherein they come up throughout the story) and you the player can agree with him as much (or as little) as you like. In that scene, going by the text only, Cloud shifts the focus away from the disaster, towards finding the kidnapped Aeris, and to unravelling the mystery of the Ancients. It doesn’t mean he doesn’t care, but as to whether or not he does, that’s largely up to the role-player.
I felt like he definitely cared and was definitely hurt, but was trying to be strong because he knew he still had work to do and Aerith and Marlene were still out there. It’s hard to portray emotions through text alone, but I expect the remake will really open that scene up some more.