[after your edit, and all other answers] Thanks. Now I understood your point (I think).

By the way, I notice that X->SpecZ is not a model of X->SpecC over Z !

------------------ So, now I'm lead to think that a structure of scheme over Spec(C) plays no role at all, since an integer point is a Z-valued point OF X_Z, NOT OF X ITSELF...

I mean: the line {x=pi}, being isomorphic to SpecC[t] (even over C), as an abstract scheme will have (several?) models over Z, but none that is induced by THAT inclusion in A^2_C. Right?

The two lines {x=0} and {x=pi} are schemes which are isomorphic over SpecC, hence they are a fortiori isomorphic as bare schemes over SpecZ. When you say "the line X'={x=pi} is not defined over Z" you really mean: there is a canonical (or, rather, standard) model over Z of the ambient scheme A^2_C (just the one induced by the inclusion of rings Z[s,t]->C[s,t]) and the sheaf of ideals defining X' is NOT the pullback of a sheaf of ideals on A^1_Z. Is that right?

@Geoorges: Do you mean that the notion of "integer point" of a scheme is not intrinsic at all but depends on a choice of "coordinates" (embedding in affine/proj space...)? [BTW, which is the correct formal definition of "integer point of X"?]

So, my question can be rephrased as: what's the relation between the notion of integer point of X (whether it is intrinsic or not) and the notion of string-of-integers-that-solve-a-given-equation-for-X-in-affi‌​ne-space ?

Dear Georges, the first paragraph of your answer just repeates what I asked in the "very very naive question": having integer coordinates is a coordinate-dependent notion, not an intrinsic one.

@Q.Y.: So? A morphism from specZ to X would of course be a morphism of schemes, not of schemes-over-C. What I was asking is whether there is -in any sense- an interplay between considering "further structure" on X and the notion of "integer point". [BTW, what's the correct definition of integer point? ]

Muthiah was right, it's easy: just take the diagonal inside the product of the two open affines etc.

Ok, found! It is spelled out in Mumford's book. Before I was only looking in Hartshorne's.