Apparently you are talking about something that no one actually calls consolidation final, but what we call a 3rd/4th place decider.
Consolidation final is only used as a term for a match in a double elimination bracket, where a 3rd place playoff match doesn't exist.
You don't win anything to become 3rd, you lose the consolidation final, so this is not a "third place playoff" match.
You've obviously got something mixed up by even bringing the 4th place into this discussion, since 4th place is decided before this match, in the lower bracket final.
A 3rd/4th place match in a single elimination bracket can be called a consolation final, that would be fine, and that is what your wiki link is about. We don't call it that around here, but we do use 3rd/4th place decider or a bronze medal match.