Two different objects
One is a property of the method. The other is a property of a sample.
Edge is a characteristic of a trading method: the expected value it produces per unit of risk, net of costs, over the population of trades it will generate. Profit is a characteristic of one finite, path-dependent sample drawn from that population. They are different kinds of object, and the relationship between them is statistical rather than definitional — which is precisely why a method with edge can produce a losing quarter and a method with none can produce a winning one without either outcome being surprising.

