Question

what is the difference between thinking of a person as being employed versus thinking of person's labor as being employed?

Answer 1

Obviously, the two concepts are very close to each other. A subtle difference can arise due to different analyses. When we think of production potential or actual production, we generally want a measure of the total labor being used (employed). For example, a firm's production function is often described as Y=f(K,L) where K is some measure of capital and L is some measure of labor. Generally, we assume that L is being used efficiently.
When we talk about the unemployment rate, for example, we are counting people (sometimes households). Yes/no are you currently working. There is no distinction between working at ones highest potential or not.

I hope this helps.