To friend vs. to befriend

I overheard a mother correct her child who said something to the effect of

There was a new kid in my class today and I friended him.

The mother said befriended and I was inclined to mentally agree (given the kid was maybe six or seven, it’s unlikely that he was referring to social networking).

However, with a little Googling, I discovered an article from The Atlantic which suggests that ‘Friend,’ as a Verb, Is 800 Years Old.

• “Make no purses, for to friend yourself therewith.” — the Guide for Anchoresses, early 13th century
• “And after soon friended were the King David of Scotland and Stephen, king then of England.” — Andrew of Wyntoun, Chronicles, c. 1425

This answer regarding the be prefix offers:

The general form is:

   be + [quality]

and the corresponding meaning is:

 to turn into + [quality].

So, I guess my question, then, is what is the difference between the verbs to friend and to befriend? Was the kid right?