This definition of messenger is debatable and has issues with it.
To answer your question though, we know for a fact that Adam (AS) was not a messenger without resorting to any theorized definitions. It comes in a hadith regarding the Day of Judgement:
Allah will gather the believers on the Day of Resurrection in the same way (as they are gathered in this life), and they will say, 'Let us ask someone to intercede for us with our Lord that He may relieve us from this place of ours.'
Then they will go to Adam and say, 'O Adam! Don't you see the people (people's condition)? Allah created you with His Own Hands and ordered His angels to prostrate before you, and taught you the names of all the things. Please intercede for us with our Lord so that He may relieve us from this place of ours.'
Adam will say, 'I am not fit for this undertaking' and mention to them the mistakes he had committed, and add, 'But you'd better go to Noah as he was the first messenger sent by Allah to the people of the Earth.'
They will go to Noah who will reply, 'I am not fit for this undertaking,' [...] (Very famous hadith recorded in many books; this is a version from Sahih Bukhari)
In this hadith, Adam (AS) himself calls Nuh (AS) the first messenger for mankind. So, it is clear Adam (AS) was not a messenger as he came before Nuh (AS).