A k-variety is called a variety if it is absolutely irreducible, i.e. is not the union of two strictly smaller kalg-algebraic sets.
One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine n-space.
The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set up to change of basis.
The absolute Galois group Gal(kalg/k) of k naturally acts on the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, ..., xn].
In the affine case, this means the action of the absolute Galois group on the zero-locus is sufficient to recover the subset of k[x1, ..., xn] consisting of vanishing polynomials.
In general, this information is not sufficient to recover V. In the example of the zero-locus of x1p- t in (Fp(t))alg, the variety consists of a single point and so the action of the absolute Galois group cannot distinguish whether the ideal of vanishing polynomials was generated by x1 - t1/p, by x1p- t, or, indeed, by x1 - t1/p raised to some other power of p. For any subfield L of kalg and any L-variety V, an automorphism σ of kalg will map V isomorphically onto a σ(L)-variety.