I interpret work to live as the reason I work is to live. That implies working is necessary to live. Which is simply not true in our modern day society. Some people don’t have to work.

I could see how someone could easily misinterpret “work to live” as deriving their sense of self of life’s purpose from their job. That is the opposite of what is meant by it. It is so close semantically to the exact opposite philosophy of “living to work”.

Working to survive, on the other hand, implies that your only there because you are forced to be to survive.