The truth is, at the below $2,000 price range, those integrated amps are not going to be distortion champions as they don't typically use the near SOTA DAC and volume control ICs, and tend to use standardized parts with their AVR/AVP lines. So if measured on the bench (they they have been), you will find that they don't offer less distortion. Also true, that even at their mediocre level of distortions, you won't be able to tell anyway, only the so called golden ears will tell you they hear better sound quality.
So, typical answer could be, pick the one that has the features you like better, and/or looks better to you, or based on the best prices you can find for each. If it is my money, I would skip integrated amps and go separates because I am ocd on distortions, like you said, the less the better regardless.