You are not considered to have leave to remain permission until you have registered with GNIB. Your first stamp starts at the date of your registration.
The only case where a person's residency can be traced back to the date of arrival is refugee. It will be specifically spelled out on INIS and Citizen Information website.
A person granted refugee status can apply for citizenship through naturalisation once they have acquired 3 years of reckonable residency. Residency is calculated from the date of arrival in the State.