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Let’s say I have a 2010 13" Macbook with a dead logic board. And let’s say I have access to (friend has one) a 2010 13" macbook pro that works. If I want to test the display from the Macbook, is the logic board display connector the same on the Macbook Pro? Just trying to find out what’s still good on the Macbook without getting an entirely new machine, if it can be helped.

No these are different display panels. White macbook have a cable inbuilt with camera also means camera and display connected on same cable but in MacBook Pro 2010 have a different camera cable and different display cable . So you cannot try it . it may be gives you a problem.