There's an element of truth in both definitions, and they're both flawed. 'Professor' isn't a qualification; it's a position. US universities are more inclined to give people such status - it's all part of the typical status inflation that affects the modern world. (Some companies have 30 or 40 Vice-Presidents). I expect part of the pressure for it is due to the melting pot of language among US immigrants; in many European languages the prof- word just means teacher.
Autobigraphical tid-bit: I was taught Romance Philology by one Dr Cremona. He was a leading authority, widely published, and by far the most knowledgeable in this area on the staff of the University. But the 'Professor of Romance Philology' was a man who was a distinguished classicist - not a philologist, and retired from any teaching duties; so Jo (my teacher) was not a Professor. In contrast, a post-graduate student in a US university will have the job position 'Assistant Professor' and expect to be addressed as 'Professor'.
As a rule of thumb, call someone 'Professor' if you know they are. In British universities, most holders of PhDs (or DPhils) are not Professors, and should be addressed as 'Doctor'