Baba is Shirdi Sai Baba; the Sakthi or glory of both is abheda, 'without any difference'. He is the same magnificence that blessed, consoled, encouraged and saved, while at Dwarakamayi at Shirdi.
'Sathya' Sai Baba refers to Himself as only, 'Sai Baba'. Even now millions know him only as Sai Baba. 'Sathya' was His name, before the announcement and it was added only later by some who attempted to distinguish the 'Abheda Sakthi'. "Sathya" Sai Baba can be taken to mean that at Puttaparthi we have the Sathya (True) Sai Baba!
Baba creates and gifts to persons, pictures or images of Sai Baba or Himself, without any distinction. Worship offered to Sai Baba at Shirdi reaches Him, He declares. The words, assurances, actions, miracles, and even gestures and mannerisms of Sathya Sai Baba correspond to those of Sai Baba of Shirdi. So this is an Abheda-avaathara, with the Sakthi, identical and undiminished.
All who resort to the shrine at Shirdi and pray are really praying to Him. He knows their needs and He grants them the boons they desire and deserve. When an aged devotee conducting regular ritual worship at Shirdi Sai Baba's image came to Him, He said, "I like the sincerity of your worship, but why do you ask men to help you? Ask Me."