maybe it uses the g word so much BECAUSE it’s in the constitution…

I expect they co-authored the constitution and other prior 'foundational documents' with Claude, so it's probably a chicken-and-egg thing.

I believe the constitution is part of its training data, and as such its impact should be consistent across different applications (eg Claude Code vs Claude Desktop).

I, too, notice a lot of differences in style between these two applications, so it may very well be due to the system prompt.