Let's see if this anecdotal illustration better explains how I see it:
I live on planet X (not a real planet we know) where gravity's proportional G is ten times what we know here as G, so Xian's gravity is 10G (in Earth terms). I very carefully measure this Gx, set up my weights of measure in kilograms per this Gx, then work out the Equivalence Principle per F = Ma = G*(Mm)/ r^2 (everyone knows gravity and inertial mass are related), so my mass Mx (and mx) is measured in the kilograms I developed. Now I am content, since I worked it all out, where my units of measure for weight on X are measured in kilograms, for which I then established an equivalence with F = Ma, where Mx is measured in kgx. Confident, I now teach at a prominent Xian university and (since I never traveled off world) merrily accept that my Gx and kilograms kgx are universal.
Four hundred years go by and in a very fancy space ship arrive people (to every Xian's surprise) who say they're from some far off place called Earth. Now these Earthians (all descendents of a prominent university where physics had been taught with confidence for the past 400 years) are very eager to impress their newfound Xians, so they too go and measure G and kilograms. To their surprise, they discover that the Xians are using a different unit of measure for kilograms than the Earthian measure. So they carefully explain to the Xians that G is not what they thought it is (since it is universal), but it is 10 times less, and that only the "acceleration" derived from the greater gravity of their planet is 10 times greater. Kilograms cannot change. They further explain that what they had done wrong was make a tenfold mistake (or hundredfold?) in estimating their kilograms. They made the error of thinking that their Gx (which is 10G in Earthian terms) is the correct G, so the kilograms they developed was based on this error. Since, as your Earthian student descendents take great pains to explain, only "aceleration" is ten times what it should be, so mass measured in kilograms has to be the same, so their Xian kilograms are obviously wrong. The Xians challenge this, saying no, that the Earthian kilograms are wrong, because they are only a tenth of what they should be for G, as everyone who studied at the Xian universsity can tell you, and that their kilograms are correct, since the acceleration works out exactly for their equivalence principle F = Ma. In fact, they (barely) tolerantly explain, the Earthians had got it wrong. In thinking that G is only a tenth of Gx (everyone knows this is a universal constant), it is the Earthians who should adjust their kilograms to reflect the correct G. And that correct measure of mass is tenfold Earthian kilograms, so obviously Xian kilograms are the correct measure for mass.
Well, this heated discussion goes on for some time, and as their appears no solution to this problem, with both Earthians and Xians convinced their measures for mass are correct, it appears the two worlds are in danger of declaring war. Knowing that we really don't know, my (long descendent) student politely (diplomatically) reminds his friends at the Xian academy that it was smart Earthians who came to Xian, and not the otherway around, so perhaps Earthian measured kilograms of mass should be adopted (though they are different from Xian kilograms), just to keep the peace. But he gets shouted down because they say the Earthians are on X now, so must use Xian kilograms instead. :-?
Who is right?