Yup, DE is double ended and describes any metal halide that has double ended sockets (small bulbs).
Moguls are the common single-ended sockets used, around the size of regular light bulbs you use in the home (different socket though, so don't try a MH in one

)
There's also a medium socket that is single-ended and smaller in size then the moguls. Mediums are much more rare.
HQI is a term that was originally patented by Osram of Europe to describe a specific type of MH technology, but it has since been used to descibe anything that is DE. Most of the "HQI" systems aren't true HQI, but the moniker has stuck and so everyone assumes HQI just means DE.
Pulse start is a specific type of ballast technology that uses an ignitor + ballast to start a bulb. They're all designated ANSI number M13x (replace x with different numbers for different wattages). Most MH sold are PROBE-start (ANSI M5x series), but some have suggested pulse-start is the far better way to light certain bulbs. For example, european/german bulbs usually don't have a starting probe in the bulb so they really should be operated with pulse-start ballasts. However, most people use standard probe-start ballasts to fire them up. It works, but just not as well.
Lighting is REALLY confusing, I know. Let me know if you have any more questions.