I found these two paintings of the ships. In one it clearly shows the flag of Castile and Leon, but I have no idea which flag could that be in the other painting. What could be the reason for this discrepancy?
Columbus carried two different flags. On board of the Santa María he had the royal standard, and the three ships carried flags with a green cross and the letters F and Y. I'll quote Columbus himself on the matter, from the diary of the first voyage, copied by Bartolomé de las Casas:
Luego vinieron gente desnuda, y el Almirante salió a tierra en la barca armada, y Martín Alonso Pinzón y Vicente Yáñez, su hermano, que era capitán de la Niña. Sacó el Almirante la bandera real y los capitanes con dos banderas de la Cruz Verde, que llevaba el Almirante en todos los navíos por seña, con una F y una Y: encima de cada letra su corona, una de un cabo de la cruz y otra de otro.
Translation: Then they saw naked people, and the Admiral went to the land with the armed barge, and Martín Alonso and Vicente Yáñez, his brother, who was captain of the Niña. The Admiral pulled out the royal banner, and the captains two flags with a green cross, that the Admiral carried in all the ships as ensign, with an F and a Y, and above each letter its crown, one on one side of the cross, and the other one in the other.
This is corroborated by the latter the Admiral wrote to Luis de Santángel, that mentions the royal standard: Sir: For I know that you shall find pleasure in the great victory that Our Lord has given me in my voyage, I write to you this letter, through which you shall know how in twenty days I reached the Indies with the fleet that the very illustrious King and Queen our lords gave me, where I found many islands populated by innumerable people, and I have taken possession of all of them in the name of Their Highnesses, with public proclamation and flying royal banner, and it was not contradicted.
What did the royal banner look like? It's a bit hard to tell, but it probably was not the banner of Castile and León, but rather a combination of the arms of Castile-León and Aragón-Sicily, as prescribed by the protocol on the use of the coats of arms signed in the Concord of Segovia of 1480. This point is further proven by one such banner surviving in Granada from the time of the Catholic Monarchs. You can see it in this brilliant report by El Independiente de Granada