I find it confusing to call this kind of attack "SQL injection", as it is much better described by "privilege escalation", isn't it?

I wouldn't say that. Elevation could be done in a myriad of ways, and injection is one of them. The question is whether you are causing a query to go wrong by feeding it specially crafted text. The only scenario without useful injection is one where you have complete and convenient root access from all input methods.