I have heard that Athena was born out of thoughts, sprang from Zeus' head. In that case, does this mean Athena is a manifestation of Zeus' intelligence?
The Metis story is interesting because according to the prophecy Metis would have two children - Athena and then a son who would ovethrow Zeus. This would have represented a triumph for the Titans, including Metis, over Zeus and his Olympians. So Zeus prevented this next generation from being born by swallowing Metis whole, thus absorbing her wisdom as well.
So we have to assume that Athena has the prudence and forethought of her mother, and the cunning of her father. (Metis had cunning too, as she came up with the plan to free the Olympians by making Kronos vomit them up.)
Yes and no. It depends on how you look at it.
Athena is Zeus' favorite. She is a "daddy's girl" for sure, siding with the father over the mother in the trial of Orestes. He gives her the aegis to make sure she is protected in battle, and the term "aegis" is still used to connote a form patronage or protection.
In this light, her wisdom, and her springing fully formed from the head of "scheming Zeus", can be seen as manifestations of Zeus' intellect.
By contrast, Zeus swallowed Metis to eliminate the threat of being overthrown by his own offspring, presumably wiser than himself, and in that context the story may be read as Zeus transforming Metis' wisdom into a governable form, so that it may not be used against him.
Antithetical to Ares, Athena (equally patron deity of warfare) is slow to anger. She is not impulsive, brash, impudent. She is prudent, indeed wisdom incarnate. She's not born of the carnal impulse. Rather than think of Athena as an aspect of Zeus, it might be more fruitful to consider each deity an anthropomorphic aspect of the human experience. She represents our faculty of wisdom in action, probably best expressed as "prudence". This faculty not only wins wars, it ensures that warfare is the correct course to begin with, i.e., that the war is just. It is not without significance that Athena is the patron deity of Greece's capital city: what better than the principle of prudent protection to oversee a city? There is also a fascinating interplay between she and whom she bests to become patron deity of Athens - Poseidon - since each represents a gender inversion of typical symbology. Athena, though female, represents the masculine principle of combat, and Poseidon, though male, represents the oceanic expanse.