As shown in the below screenshots, when placing water on the backside of stairs, the water flows through to the other side of the stairs rather than where it's intended to be. This happens both in creative and survival modes with an individual stair placed and when it's connected to other stairs (i.e. making a well). However, when placing lava, it works as I'd expect it to.
I've searched for "Water, Stairs" and the closest ticket similar to this issue is closed as "Incomplete" (MCPE-55588). I haven't been able to find another ticket explaining what I'm experiencing.
According to the 1.4 changelog:
So, stairs will block flowing water, but if placed in source water or source water is placed in them with a bucket, they will behave as both stairs and source water blocks. (You can reading more about waterlogging on the wiki.) Stairs, uniquely as far as I know, do not allow water to flow out of their full sides, so they can be used to direct water flow from a source. This is a feature, not a bug.
If you want to fill the enclosure in your pictures with water, try using the bucket on the ground surface behind the stair, rather than on the back of the stair.