How can I complete my derivation of an algorithm to solve Sudoku puzzles from my formalisation of the rules?
One method of dealing with these, somewhat, emergent rules is constraint solving. You can iterate through each cell and place constraints on other squares based on that cell.

For example, if a square is 3, you can place a "cannot be 3" constraint on all other squares in its row, column, and 3x3 box. The two examples you provided can be solved once you place all the constraints, and look at what options are available.

This sort of method is very similar to what you would do when solving a sudoku puzzle when you write down the possible numbers in each square, and fill a box in when there is only one possibility.