Oh, I see that Elizabeth ("Lily") has a sister Ethel May Spence who is with the Berry family, age 42, on the 1920 US census, and there is an Ethel May Spence birth registration in 1877 in Kirkstall, so it looks as though you should have found the right birth registration for Elizabeth. There is a likely family on the 1881 England census with parents George and Elizabeth and children including Elizabeth 6 and Ethel May 3, and their birthplace is given as Headingley.
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