a lattice point
A lattice point is one dot in that infinite grid — one address in the periodic pattern. Its defining feature: it has exactly the same environment (the same neighbours, at the same distances and in the same directions) as every other lattice point. If two points do not have identical surroundings, they cannot both be lattice points.
Lattice points are abstract positions, not atoms. In simple metals an atom happens to sit on every lattice point; but in general a point merely marks where the motif is anchored, and that anchor may fall on an atom or between atoms. For example, in a body-centered lattice the corner point and the centre point are equivalent lattice points — each is surrounded identically by 8 nearest neighbours.
A common error is to call every atom a lattice point. In NaCl, Na and Cl are different atoms with different environments, so they cannot both be lattice points; the lattice point marks the Na-plus-Cl pair (the motif). The test is: does this position have an identical environment to that one? Only if yes is it a lattice point.
In body-centered cubic iron, is the atom at the cube centre a lattice point? Yes — pick it as a new origin and the eight corner atoms sit exactly where the centre atom's neighbours sat from the old origin. Corner and centre are indistinguishable, so both are lattice points; the lattice has 2 points per cell.
Corner and body-centre atoms in BCC are equivalent lattice points.
The number of lattice points and the number of atoms are different counts: one lattice point may carry a motif of many atoms, so atoms per cell can far exceed lattice points per cell.