Yep, that's generally true. Trance though is slower (130-140 bpm most of the time) and very melodic.
That remix probably qualifies as happy hardcore, what with the piano & poppy synths. It's kind of a combo of techno & trance, only faster and with happier synths than normal hardcore.
I will admit, some genres seem indistinguishable to me, and most are just combinations of other genres.