Medium Killer Sudoku
A medium killer sudoku is one that stops rewarding arithmetic on its own. The cage totals carry you some distance and then the board goes quiet, and the move that restarts it is not a harder sum. It is a change in the question you are putting to a row.
What the label is measuring
Searches for killer sudoku medium land on a great many pages where the word means whatever the site felt like that morning. Here it is measured. Every puzzle is solved before you see it, using the techniques in the order a person learns them and always taking the easiest move available, and whatever the puzzle forced becomes its grade. A board arrives here having been forced into a hidden single at least once and never past one: cage arithmetic and lone candidates were not enough on their own, and nothing above the hidden single was ever called for.
The definition is narrow on purpose. Counting cages would be simpler and would tell you nothing, because two boards with the same number of cages can be a long way apart depending on where those cages sit.
Naked and hidden are opposite questions
A naked single asks about a cell: this square has one candidate left, so that is what goes in it. A hidden single asks about a digit: this 7 has only one square left in the row that will accept it, so that is where the 7 goes, even if the square in question still has five other candidates of its own.
The second question is the one beginners forget to ask, because the cell does not look finished. It looks wide open. Nothing about that cell tells you the answer; the information is sitting in the eight other cells that have ruled the digit out.
Why hidden singles fire so often on a caged grid
On a plain sudoku, candidates are eliminated by the digits already printed, so early in the solve there is not much to work with. On a killer grid the elimination begins before you have written a thing. A cage totalling 6 across three cells can only be 1, 2 and 3, which strips six digits out of each of those cells immediately, and those strippings ripple along the rows and columns they touch. Hidden singles then start turning up in places that would still be featureless on an ordinary board.
Which is the argument for doing the cage arithmetic thoroughly rather than opportunistically. It is not merely a way to fill a few cells early. It is the thing that makes the rest of the grid readable at all.
A working order
Turn Notes on and pencil each cage’s possible digits into its cells before you place anything. Fill the cages that admit only one combination. Then walk the nine rows, the nine columns and the nine boxes, asking of each digit in turn whether it has been reduced to a single home, and place it when it has. Repeat until nothing moves. On a medium board that loop generally finishes the puzzle without any technique beyond those two singles, which is exactly what the label is promising.