Ha cai (Cantonese) or yu tsoi/tsai (Mandarin) is pretty much the same as what Americans/Canadians call "rape" -- it's the plant that canola oil comes from, Brassica napus. There are several cultivars, but they all taste the same as far as I can tell. The Chinese name for the plant means "oil green-leafed-vegetable," which pretty much says it all.

Gai lan is Brassica oleracea -- they're related but not the same, as you guessed. Gai lan is closer to cai soong/choy sum in taste. Gai lan is closer to broccoli, cai soong is closer to mustard greens, or so I think (I'm sure there are people who would disagree with that though).

But I think your thoughts about oyster sauce are mmmmmmmmmmm. I haven't yet met the brassica that didn't get along well with oyster sauce.