Calculus 3 in the U.S. is multi-variable calculus. Sometimes and introduction to linear algebra is given in this course. Calculus 4 is differential equations. It’s almost never officially called Calculus 4. Sometimes an introduction to linear algebra is given in this course.
Okay that makes sense, that's how the classes went for me in the US, with Calc 3 being multi variable (and as you say, a brush with linear alg via vector-valued functions), and 2 sideways steps from there, one being a course on PDEs and the other on linear algebra, both called by those names rather than listed as a step in the calculus sequence itself.
