If you are using width to stand in for height - the height can't move very much, that's just the size of the tire circumference when it's all strained out. It's probably nearly the same in spite of rim width. But the width at the widest point is definitely being taken in a lot by skinnier rims. Paselas are 33 mm wide on my Paramount but the same ones are 29 mm wide on these skinny Rigidas

That said the Serfas I pulled off the Rigidas measured 27mm, 2mm narrower than the Paselas despite being the same listed size and a lot thicker carcass