This is mostly my opinion and theory on the promise shared between Cloud and Aerith.
Cloud and Aerith may not have sat on a well and looked up at the starry night and made a promise. But with clues like the LOVELESS play, and the Promised Land, it gives off the idea they shared a connection, that was only special to them. The LOVELESS play symbolizes their love, and even quotes sound like they're Cloud and Aerith's. At the ending of the game, Cloud says he wants to meet Aerith in the Promised Land. And earlier, Aerith said she wanted to "meet" Cloud, and maybe she would get that chance to in the Promised Land. It's sorta like the promise between Squall x Rinoa. They would meet in a place they could be together. The Promised Land is that place for Cloud and Aerith.