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'
Student or Learner