Please don't take everything you find on Family Search as absolute truth. Some of it is from submitted records, and these are based on other peoples' conclusions, which may be wrong. We have already seen conflicting information from the census, so I wouldn't be sure about her place of birth or her parentage until you see a birth record.
If her age on the 1900 census is accurate, Elizabeth would have been approximately age 43 at the time of Lillie's birth. She & James were apparently married for many years with no offspring, so that makes it seems even less likely that she'd have been able to bear a child at that age. I really think it's likely that Lillie was adopted. Italy might have been listed as a birthplace because she had Italian heritage & the person giving the information was confused.
James appears to be listed on the 1920 Census in Kansas City, Missouri as father-in-law to Carl B. Mears:
https://familysearch.org/pal:/MM9.1.1/M8HJ-YKP Lillian Mears is listed as born in Missouri & parents born in England.