In your top condition, considering the rolled corners of an HSS, you`d be using flare groove welds.
For your bottom condition, you`d have a fillet on the bottom, a butt weld (which is a non structural weld) on the sides. I`m not sure what I`d call the top.
I wouldn't expect much measurable/reliable capacity from this set up, and would suggest you get into beveling the edges and using a partial penetration weld instead.