"The birds flew over the lake" is the better sentence because the preposition over is used to say that something is moving across, or from one side to the other, of something else. Above is not used in this way.
The prepositions over and above both tell us where something is in space with respect to something else, but they are not always used in the same way.
Over is used to say that something goes from one side to the other side of something else, or upward and across something. It tells us that something is moving from one place to another.
Above is used to say that the position of something is in a higher place than something else.
Sometimes over is used to mean above when you are saying that something is in a higher position than something else.
Above is never used to mean over when you are talking about going from one side to the other side of something. Above is not used to say that something is moving from one place to another.
When choosing the correct preposition, use over if something is moving across or from one side to the other of something else, and use over or above if something is in a higher place than something else.
I hope this helps.