The Empadronamiento (Padron) is the register that the local council hold. You should register on this to ensure that the council gets funding for local services from central government.
The
Police come and check that you are there, and they sometimes do a revisit to check.
The Padron for travel is for residents to receive discount when they travel to other islands or to mainland Spain. This has to be updated at the council office in Antigua/Caleta. It costs 50 cents. I believe that in La Oliva it can be renewed online.
The Residencia certificate/card is different. You have to apply through the
Police Station in Puerto del Rosario. It proves that you live here full-time.
The residencia has to be shown to get the 30% discount card for travelling on the
buses.
It's not possible to re-charge any of the
bus cards online. It has to be done through the
bus driver. The 5%
bus card is transferable, so anyone can use it and there is no time limit on it. The 30% discount card has a photo of the holder and is not transferable.