Many people call their children a different, diminutive name day to day, to be more informal. "Thomas, Tom/Tommy", "Henry/Harry", "James/Jamie/Jim", "William/Will/Willy/Bill"
Maybe they recognised that Lilibet was particularly special to the queen and while they wanted to pay that tribute to her, they knew it wouldn't be "seemly" to call her the exact name day in, day out.