What's the term for moving/promoting code between deployment environments
I don't think there is a generic term for promoting to production. I can tell you that within the devops community, terms are frequently interchanged. As long as the point being addressed is well thought out and made clear, the point can be understood. Some of the terms that are used when referring to the movement between environments are deploy, propagate, move, promote, and release. I will say though, whenever I hear the term stage, I usually associate it with a pre-production environment. I think most others would assume the same. 

Within your organization/groups, I think it is important to try to be consistent with your terminology. This eliminates any confusion and assumptions people can make. An example would be, in my organization, we use the term deploy whenever we are referencing moving between environments except when referring to the production environment which we use the term release. This is to emphasize that it is the final movement in the pipeline/process.