I fear, the explanations are a bit confusing.
There are essentially two points:
- zener diodes are designed to achieve a stable, mostly temperature independent breakdown voltage. Diode current depends on the external circuit, continuous and pulse power ratings have to be observed
- a "regular diode" has no tightly specified breakdown voltage and neither a defined breakdown characteristic. The actual characteristic is usually less steep than of a zener diode. In addition, breakdown may occur only for part of the diode chip, e.g. at the edge, causing local overheating and damage at relative low reverse current.