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Which expression defines ppm?

(grams analyte/grams sample) x 10^6

Parts per million expresses a very small concentration as a ratio of the amount of solute to the total amount of the sample, multiplied by one million. In other words, ppm = (mass of analyte)/(mass of sample) × 10^6. That mass–mass ratio is what defines ppm, so using grams of analyte over grams of sample (and then multiplying by 10^6) directly matches the standard definition.

The other forms mix different quantities—moles per liter is a molar concentration, not a mass ratio; a volume-to-volume ratio isn’t a mass ratio; and grams per liter is a mass-per-volume term that isn’t the canonical mass-to-mass ppm unless you convert to a mass ratio and apply a density-based adjustment. In dilute aqueous contexts, ppm often correlates with mg per liter, but the strict definition is mass of solute per mass of solution.

So the expression using grams analyte over grams sample times 10^6 is the correct way to define ppm.

(moles analyte/liter of solution) x 10^6

(volume solute/volume solution) x 10^6

(grams analyte/liter of solution) x 10^6

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