The birth registration I found would be the correct one then, as it doesn't refer to the date of birth, but when the birth was registered. As she was born June 18th, she probably wouldn't be registered until beginning of July, which would put her registration in the Sept quarter. The actual certificate would show the exact date of birth.
Eva Lily Shirley age 11 on 1901 census looks a very strong possibility for her mother. She is the only Eva Lily Shirley to be found in Staffordshire of about the correct age to be your Nan's Mum.
If you sent for Eva Lily's marriage certificate, that would give you her father's name (which will most probably be Thomas leonard Shirley.
Would you like any further look ups on this line?
Suz