Powered and amplified are two different things. An OTA tuner is designed to detect a signal with a certain minimum strength/signal-to-noise ratio. All tuners have some form of internal amplification, it's part of the tuner circuitry. That's not the same as using an external preamplifier or amplifier, whether it's built in to the OTA antenna or not. External preamps need some sort of power. Sometimes that comes from a satellite cable that is routed through the OTA antenna (won't work with DirecTVs new satellites). If there's no power from a satellite cable, then you need to send power up the cable connecting the OTA tuner to the OTA antenna/preamp. The HR20 does NOT have power on its OTA output so you have to use a power injector.
And according to the threads I just looked at, the Dish 622 does not have power on the OTA cable either but the Dish 942 does. See this thread
http://www.satelliteguys.us/dish-hd-discussions/69834-dish-dvr-622-ota-problems.html