Townsend includes those, but also draws upon Nahuatl accounts written shortly after the conquest. Those sources haven't been paid enough attention.
What's interesting is how highly he speaks of his indigenous allies and even opponents, there's not a hint of disrespect (well, minus the Aztec practice of human sacrifices of course), he praises the skill, courage and discipline of the warriors, the craftmanship of their weapons and armor, and he's full of awe and wonder when they first arrive in Tenochtitlan:
"Among us were soldiers who had been in many parts of the world, including Constantinople, all over Italy, and Rome, and they said that they had never seen a plaza so well laid out, so orderly, so large, and so full of people."