This page of birth registrations was all births elsewhere but registered in Providence.
There were many opportunities during this era for work with the textile mills, clothing manfacture, machine shops, jewelry & other industries so some people seemed to move around a bit.
I theorize, based on my family's experience, the reason for the delayed registration taking place in Providence rather than where the birth actually occurred was that the family was in the process of relocating to Providence when a birth occurred & waited until settling in their new hometown to register the birth.
With my family the birth occurred in France during the family's journey to the US & they waited until they got to their new home of Providence before registering the birth. The birth was in Dec 1892 but the registration wasn't done in Providence until the end of April 1893.
Your family may have been in Providence briefly between censuses & so registered the birth in Providence but then left for a work opportunity in Lawrence.
I've found though, with the censuses, birth place isn't always accurate. The answer all depends on who was supplying the info to the enumerator & sometimes in multi-family structures, that person wasn't always a member of the familky but a neighbor or boarder in the family household. I've one 2x great grandmother b. NY, MD or VA depending on the census one looks at. NY is the correct answer.