Do the words cleaning and tidying translate differently in different English speaking countries? Specifically, would vacuuming always be considered part of tidying AND cleaning?
Well put it this way. The verb 'to clean' is different to 'to tidy'.
They can overlap in the context of say, 'cleaning/tidying your room' implying that a mess needs to be cleared.
Another contextual situation where they don't overlap would be 'to clean the mud on floor' or to 'tidy the books on the shelf'.
You wouldn't 'tidy the mud on the floor' nor would you 'clean the books on the shelf' (unless the books were really dusty, but then you might use the verb 'to dust')!
Context is important, they can be interchanged but only where it makes sense. Tidy generally implies sorting whereas clean generally implies washing or cleansing.
If my wife was leaving for a few hours and she told me to "really clean up the living room" that would imply, in addition to picking up the scattered newspapers and day-old pizza boxes, to vacuum the wall-to-wall carpeting.
However, if we lived in Florida instead of Minnesota, the same command would suggest wet mopping the ceramic tile floor rather than vacuuming. (Of course, whether either got done without her standing there watching me would be a good question. After all, I have lots of questions to answer on Stack Exchange.)
What one does to "clean" is highly dependent on ones situation. Not only are there differences in the physical surfaces to be cleaned, there are also differences in the assumed starting state (is it one layer of pizza boxes or two?) and differences in the expected final outcome (did she really expect me to dust that old vase?).
So terms used for household cleaning, if not voiced with specificity, are highly dependent on the particular situation. About all one can say is that "clean" involves removing dirt/dust (from somewhere), while "tidy" suggests removing old newspapers, etc, and placing any displaced furniture and knick-knacks in their "proper locations" (but whether newspapers are merely stacked on the floor beside the couch or totally removed is highly variable, as is how specific "proper location" is).