I was using skinnier as a one-word shorthand for ‘less heavy’, but you're right that a volume-based definition is closer to the common understanding than a mass-based one. (Cf massive which is also about mass in technical speech but about size in colloquial speech, though for a different reason.)

(Plus, in most cases of people trying to lose weight, they would actually care more about fat mass than total mass if they fully understood the difference and could measure both.)

(In Italian we have a phrase falso magro lit. ‘false lean [person]’ for people who weigh more than one would guess by looking at them.)

Also, a person with lots of muscle definition won't look "fat" even if they weigh much more than average. They won't look skinny either, but large-and-muscular is generally considered healthier and more attractive than large-and-flabby.

I don't understand how eliminating fat in this scenario merely makes me merely "look skinnier" rather than actually being skinnier. Constant mass + increased density = reduced volume = (in this case) skinnier... doesn't it?

The parenthetical distinction was between 'losing weight' and looking (and even being) skinnier. ie. Gained weight, lost volume and subjectively appear to have lost even more volume.