Definitions of elements, nodes, branches, loop, mesh for a network with short circuit
You are correct, the terminology isn't working very well when you have a short circuit across an element. However, I don't think the terms "branch" and "node" have a very formal, standardized definition. The textbook I use to teach does not say that the number of branches is equal to the number of nodes.

I think the root of this issue is that we usually just discard elements that are shorted before trying to analyze a circuit. So, by the time we get to the point of naming nodes and branches this conundrum has been quietly swept under the rug.