&gt; ...but capitalism describes a type of ownership arrangement (private property)

Exactly, the candy bar is person A's property that he can do with as he wishes. It is not the communal property of society at large thus the ownership arrangement is one of "private" property. More to the point person A has invested his capital to purchase that candy bar which now being his private property he then sells at a profit (thus "capitalism").

What you are describing has nothing to do with an ownership arrangement or with the concept of private property but about an employment arrangement. Your definition of socialism focusing on solely on labor owning the means of production (I'm assuming) the above meets your definition of "socialist" despite also meeting the definition of "capitalist" and of "private" rather than socially owned property. At this simplistic level the system is both because the student is both the capitalist investor and labor in one person... as is true of any sole proprietorship. 

I'll only note that I have been reliably informed by other reddit socialists that even labor owned enterprises are still capitalist if they are engaged in commodity production for a profit in a market economy rather than production to meet demand in a socialist "demand economy". I confess as a free market capitalist I honestly didn't understand the guy. I don't see the difference between between a "demand economy" and a "commodity market for profit" since the capitalist (whether he *also* provides the labor or only invests capital) can only reap a profit if the goods he produces meet a need.. the two go hand in hand and I don't think you can have one without the other. 

But I'll let the socialists argue that one out between themselves.... just realize that not all socialists agree with you.