Obsidian floors will be formed on top of all obsidian tiles you create PROVIDED the floor level is free. That level is NOT free if you have an up/down staircase there (which isn't uncommon when obsidianizing the magma above a magma flow). If you remove the staircase after obsidianization you end up with an empty space tile with an obsidian tile underneath (and you can't channel away that obsidian, removal can only be done by digging from the side). Removing the staircase before obsidianizing the tile below will provide an obsidian floor.
As others have said, casting obsidian and then channeling it away results in a bottom floor of the "natural" material in that area. This means that casting obsidian on the surface and channeling it away will produce a floor of the normal soil type (including any "contaminants" in the form of patches of stone normal to the area). Such a ground level soil floor supports vegetation, including herbs (probably saplings and trees as well, although my experiments were made on top of a built floor, so there was no room for roots). Casting obsidian above the ground level ("up in the air") and then channeling it out resulted in soil that did support grass but no herbs in my experiment. As far as I've seen, obsidianizing magma in it "natural" environment (the magma sea and magma pipes, which includes volcanos) always results in an obsidian floor when channeled away.
I'm not sure what you get if obsidianizing a stream and then channeling it away, but would assume you'd usually get soil, i.e. the same floor as was there previously. If you drain a murky pool, floor the bottom, and then remove the floor, the "murkiness" property of the pool floor goes away (e.g. allowing vegetation to grow there), and if there is some kind of property associated with a stream bed floor you might destroy that property through obsidianization and subsequent channeling, as I would expect would happen if it was done to a murky pool.
The reasons for the original questions are not clear, however. Why do you want to avoid converting a stream bed to obsidian (which I think won't happen, as per the above), i.e. why does the material matter?
Why do you care about what material a volcano floor is made of (I believe I've only seen obsidian, as per the above, but I haven't cared much)?
Magma pipes (volcanos are just magma pipes that reach the surface) have a "top level" to which it will fill the pipe up with magma. This means that if you lower the level it will fill up again, unless you prevent it from filling up. The process by which magma pipes are refilled is somewhat strange, however, as magma doesn't actually well up from below, but rather rain down from above. As far as I've seen from my experiments, magma pipes are refilled by 1/7 magma randomly forming at the pipe's top level (or the highest level reachable, if a floor blocks the level above) over tiles that are directly above a magma flow tile. From there it falls down to join the top level of the magma (or on top of the previous level, if the top one is 7/7 already). I suspect these 1/7 lots of magma can "flow" sideways while in the air, thus falling on tiles beside the one it was formed over.
The above means you can safely obsidianize the top level of a volcano and channel down into the resulting obsidian to reveal the (obsidian(?)) floor. When you channel down into this floor, however, you provide a path from the magma flow above to the level you channeled from (and, and exposed levels above that one), allowing magma to be generated and rain down above that tile.