You said that BAJATA, BAIADA and BAJADA is the same name. Among the Bajata records from Ficarazzi, Sicily I have seen the single spelling variation BAIATA.
If the family went to Malta from Spain, that would be very good to know. Do you recall where you learned that? Do you have any idea where in Spain, and perhaps when they went to Malta?
Your English is fine. And I am interested in these elements.