" This is pretty much what I wanted to write. How come productivity is measured in dollars when exchanging goods/services, but constant units when this Baumol thing comes up?"

From an economist perspective there is no difference between the two. Money is just a convenient metric to be able to compare the value of multiple goods.

You don't think of you salary as the power to buy X burgers or Y beers but really it is what it is. It is just easier to say I earn XXXX$.