PS1 version has load times before battles. Icky
The DS is definitely the difinitive version, since it's basically the SNES build with the PSX extras, plus its own unique additions, and the sound's been 100% preserved (instead of totally fucked like a lot of ports on handhelds). The optional touch interface also makes chosing commands faster if you play on Active.
The cats and the time machine didn't have anything to do with CC, unless there's some cat ending in CC that's supposed to allude to CT. I know there's a few mentions of Robo and Lucca in CC, and that Crono, Marle, and Lucca make cameos as child ghosts or...something. Plus Schala was a huge focus at the end of the game.
But since Chrono Trigger was pretty much complete, the sequel kinda fell into
dis continuity among most fans.