Maybe it's a result of spilling all those secrets to Alex Witchel in the Sunday Times magazine, but Barbara Cook has never seemed so relaxed onstage. Back at the Carlyle after a longish absence, the cabaret doyenne adopted such a casual approach that she might as well have entered in her bathrobe and slippers. Waltzing into the luxe room singing "I've Got the World on a String," she paused a while to greet Hugh Jackman (looking smashing in a black suit at theater producer Gerry Schoenfeld's table) before sidling up to the piano. Performing material by composers Harold Arlen, Wally Harper (her late longtime accompanist), and Arthur Schwartzwith a bow to Bobby Short, the Carlyle legend whose recent passing prompted her to extend her run to May 27she had to restart songs a few times to get the timing right. A slightly froggy throat required sips of water. And memory lapses resulted in a few S.O.S. calls to her new musical director, Michael Kosarin, and patter that was anything but sharp. This wasn't the polished Cook of the Gower Champion homage she performed at the Carlyle in 1999. But it's still the same singer who can take a line from Arlen's "Last Night When We Were Young" and reveal the tender anguish of a passion moving from ecstasy into memory with vocal color that's like an autumn field magically flecked with spring flowers.