How to round down negative numbers?

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When discussing question on another site, I came to a disagreement about rounding down negative numbers.

The two positions were:

  1. Rounding down is always toward negative infinity.
  2. It is ambiguous. It is defined as both rounding toward negative infinity as well as rounding toward 0.

The person with position 2, never quoted sources stating that rounding down is toward 0, and I suspect he is taking something out of context.

I have always been taught that rounding down is rounding to a smaller number (toward negative infinity), and rounding toward 0 is truncating.

Is rounding down always toward negative infinity?