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If the definition is that all statements are to have their opposite meaning then simply state:

Opposite day isn't the name given to a day when one's statements have opposite meaning.

If the definition is that all statements are to have their opposite meaning then simply state:

Opposite day isn't the name given to a day when one's statements have opposite meaning.

Maybe for clarity (to make it not just a lie on non-opposite days) you could add:

...and today isn't such a day!

How about simply:

Opposite day isn't the name given to a day when one uses the opposite word wherever possible.

If you want to be even clearer (albeit harder to follow) you could say:

Opposite day isn't the name taken from a day when one uses the opposite word wherever possible.

If the definition is that all statements are to have their opposite meaning then simply state:

Opposite day isn't the name given to a day when one's statements have opposite meaning.

How about simply:

Opposite day isn't the name given to a day when one consistently uses negated terminology.

How about simply:

Opposite day isn't the name given to a day when one uses the opposite word wherever possible.

If you want to be even clearer (albeit harder to follow) you could say:

Opposite day isn't the name taken from a day when one uses the opposite word wherever possible.