I think there are way more possibilities than you have listed, but I haven't figured out how to determine the number. Here is how I am working it.
The blocks are attracted to both the wall and the floor, and the strength of that attraction doesn't matter, so I find it helpful to recast my mental image of the problem to avoid "favoring" gravity toward the floor. Here is my mental image, preserving the colors to tie back to the original.
One way to build the staircase is to build first a single stair, then to add two blocks to create a second stair, add three more to create a third stair, then the remaining 4.
Looking at my image, that is 1!=1 way to create the first step, the block with slanted blue strips.
There are 2!=2 ways of adding the blue horizontal and green horizontal to add the second step
There are 3!=6 ways of adding the vertically striped blocks to add the third step, and
4!=24 ways to add the 4th step.
So building the staircase following this process gives 1!x2!x3!x4!=288 alternatives.
But I haven't figured out how to enumerate the possibilities that do not follow this "rule", basically where a block or blocks are added before the previous layer (in my image) has been completed.