Many cameras have on board delay, both those with SDI and HDMI eg, the sony EX-1 has an SDI out that sits after the mpeg encoder in the chain, it has more delay than it should and less picture quality than the sensor records, the panasonic AGAF100 has both SDI and HDMI and they have the same delay and quality.
The blackmagic recording devices are very low latency (even lower if you capture with the blackmagic SDK). Then there is projector latency (higher resolution means more latency), and processing latency as well.
HDMI and SDI carry industry standard signals, ie there is no real compression difference once it is in one of the SMPTE formats. How the signal gets to that state can increase delay, HDMI itself has no more delay than SDI, but many consumer cameras with HDMI out have a large delay.
In any pro system if you want a live feed you go through a mixer (good mixers have a few lines of delay only). They are actually not that expensive to use, the cheapest blackmagic HDMI/SDI mixer is under 1k and with some tinkering or third party software you can remote it via OSC. As there are now many 4k models you can pick up some bargains second hand of the earlier HD units. They do require you use a direct camera signal with no processing though, but do support keying if you want to generate an effects layer over live video with no delay.
Personally I dont like to get on stage without a mixer and that just in case fade to black emergency button, plus the no latency live feed is something I cannot let go of.
