Yes, there are lots of different names for the domain of a diagram, but "shape" is a common enough one. You can find it in standard sources, including the nLab [1] and some textbooks.
The article does mention "index[ing] category." I think terminology here is not especially universal: I am a mathematician but not a category theorist, and I have seen "shape" but not "scheme."
A nitpick, for a J-shaped diagram D, J is the "index category" or the "scheme" of D, not the "shape" like written in the article.
Yes, there are lots of different names for the domain of a diagram, but "shape" is a common enough one. You can find it in standard sources, including the nLab [1] and some textbooks.
[1] <https://ncatlab.org/nlab/show/diagram>