I can only say...The Lancs one is a very good fit for her birthdate counting back from June in 1921. There is one in Dec Q 1897, which is Elizabeth Annie, 1939 has her birthdate Oct 1897. The other is your Elizabeth, in the Mar Q 1898 as within 6 weeks to register.
There is no other of same namesake born around right time, with Wrexham registration.
I am not sure any family researcher can say with certainty what their ancestors did over hundred years ago. It could have been a short spell of work, or something not mentioned.
I know in my own FH research, there are some family stories I can confirm, others not. Also through DNA some skeletons.
Best wishes in your research

Cas