Isn't the phrase "written history" a little redundant?

No, it's not redundant. Yes, other cultures have had pictoral forms that would also be written, but - some cultures also had oral traditions of story telling and history. It increased community, and thus was considered an honor to learn from the person who passed it on as well. Many of the Native American tribes had oral history.

No. History wasn't always written down. Before we had printing press history was passed down through oral traditions. Also history can be in the form of pictures and artifacts (homes, tools, clothes, etc). These things do not require writting.

as well as drawing the distinction that many have mentioned between the ways human history is recorded the phrase written history also helps to differentiate between the history of human societies and older ("prehistory") periods in the life of the world.

No. A lot of history is oral history. Many people were not literate and passed their history on through stories.

History is simply the past. It could be audio. Video. Anything.

There is oral history and written history

there's a difference between history as the reality (what we consider and agree to be "the past") and the form or manner we investigate it or express it...

Semantics, purely semantics. All HISTORY is not written down. eg.'Personal'-history

Most, if not all of my tribe's history was oral