Why didn't Washington run for a third term?

Because he didn't want to. Frankly, he had to be persuaded to run for reelection in 1792, but his advisers made the case that the country needed another four years of the Washington administration to glue it together that little bit more. Two terms was a reasonable time to serve, and he reluctantly acquiesced.

1796, however, was a different story. George was feeling increasingly unwell, coupled with the strain of having managed the national government for eight years, with decades of public service before. The newspapers were also getting on his nerves, having grown more prolific and vituperative since his first inauguration; that very stress was fueled by members of his administration, funding party publications and urging others to follow.

He was keenly aware that every action he took would set precedent; this was true for everyone, of course, but the President is a bigger single part of government than anyone else. They had fought a long, painful, and expensive war to break away from a monarchy, and set up a very weak government in the Articles of Confederation to guard against tyranny. The weaknesses of the Articles did not negate the fears that the men at the forefront of the development of American governance had about power. So he left after two terms, and his example set up such a pressure that nobody tried for a third term until Grant in 1880, and even then that was largely because prominent Republicans favored him for the nomination, and he needed the money.

Of course, another primary reason for his retirement was the physical deterioration I mentioned before. He was sixty-four in 1796, and he had spent twenty years fighting on the American frontier before the Revolutionary War began. Washington was not in good health, particularly his teeth, which had been giving him serious pain for many years. He ended up dying in December 1799, and while his final illness owed a great deal to spending several hours a day in the snow and not keeping dry, hands up everyone who thinks being president would have been easier on him than that.

The root of his departure was at least half principle, the same principle which led him to resign as Commander-in-Chief of the Continental Army once the war was over. It would have been easy for Washington to take power, but he retired instead. He only left retirement because they asked him to be the first president, and after those two terms, he went right back to his farm. This is not to be hagiographic, or portray civil service as something one should be reluctant to take part in, but giving up power is not a very common thing.