Why is a non-rechargeable cell a "primary" cell, and a rechargeable cell "secondary"
A primary cell is a "prime mover" in old fashioned terms; namely a source of power in its own right until its chemistry is exhausted. That power was not "put into it" e.g. by charging but was inherent in the materials used in it, the same way that wood or coal are a source of heat.

A secondary cell is not (usually) a source of power in its own right : it is a store of power originally generated somewhere else. 

I would agree that the distinction is somewhat artificial and you could conceive of a cell which produced power once as a primary cell when first assembled, then be recharged from another power source and operate as a secondary cell. But I don't know of any such cells in practice.

(Likewise you can argue that wood is merely an inefficient store of solar energy rather than a "prime mover" and modern forestry is just a closed-loop process!)