In the phrase you quoted, the implication is one of using two different words for the same exact concept, implying a difference where none exists. It is not simply about disagreement, and indeed implies the opposite: you are not in actual disagreement with each other, but both parties want it to appear that they are referring to different things anyway.
In the original song, the examples that were given are all of "upper class" pronunciations vs. "common man" pronunciations. For example, at the time, an upper-class American would be more likely to say "toh-MAH-toh", to distinguish themselves from the average person that would say "toh-MAY-toh". But both are words for the same thing -- a tomato.
In modern usage, the phrase has taken on a broader meaning of selectively using vocabulary to distinguish between otherwise identical concepts. In the Time example, depending on your particular political affiliation, you are more likely to call the government subsidizing of private companies a "bailout" (if you opposed it) or a "loan" (if you support it), but ultimately you are talking about the same exact action. The Time headline is meant to be understood as reading "Some say 'loan', others say 'bailout', but the result is the same"; they are using the reader's presumed familiarity with the more general phrase for effect.
In your example from Japanese politics, you are talking about a case where an actual distinction exists between the two options. There is a real choice between two possible actions, and the words you use to describe each of those options differ because they must. This is not a case where you would likely use the "You say tomato..." phrase.
If someone who opposed a tax increase began referring to it as a "penalizing the rich", for example, then it would be more appropriate to use that particular turn of phrase.
Best Answer
Save the date or mark your calendar would be customary. Save this day would be acceptable, but not as common.
I would not use save the day, as that phrase more commonly uses a different sense of save, to rescue or safeguard. If someone were to save the day in reference to a soccer match, I would think of a player who attained victory for the team when defeat was imminent, such as scoring a last-minute goal or preventing a tying goal.