Updating multiple columns using Case
Both columns specified in the SET clause of your UPDATE are going to be touched (written to) for all the rows matching the some_more_conditions, no matter what x_specific_condition or y_specific_condition may evaluate to. There is no syntax to work around that while sticking to a single statement.

So, either live with that or go for two statements:

UPDATE some_table
SET column_x = new_value_for_x
WHERE some_more_conditions
  AND x_specific_condition;

UPDATE some_table
SET column_y = new_value_for_y
WHERE some_more_conditions
  AND y_specific_condition;