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Okay, I see, makes sense.
If we say we exchanged or swapped something, we obviously would say "exchange rate". If we say we bought or sold something, we obviously would say "price".
However, I would say property is missing in your definition:
The word "price" is the quantification of goods and services.
Since I assume you wouldn't say that people exchange dollars for houses, but they buy or rent property which means they have a price.
Now I could go into if bitcoin is property (and money at the same time) and that there are other, less convenient definitions for price but your arguments do make sense so thanks for replying.