The correct requirement of homogeneity of degree one is a little more precise than what you have cited: Extensive thermodynamic quantities are homogeneous functions of degree one only with respect to their extensive variables.

This solves the problem with your example of $U(V,T)$, but clearly reintroduces the problem of what an extensive variable is.

Personally, I do not think that an apparently more formal definition is actually much more rigorous than apparently simpler definition. The main definition is already in your original statement of size dependence. Maybe it can be sharpened a little more, but almost everything is there.

However, I think I'll add later a second answer to dig a little more on that issue.

The correct requirement of homogeneity of degree one is a little more precise than what you have cited: Extensive thermodynamic quantities are homogeneous functions of degree one only with respect to their extensive variables.

This solves the problem with your example of $U(V,T)$, but clearly reintroduces the problem of what an extensive variable is.

Personally, I do not think that an apparently more formal definition is actually much more rigorous than an apparently simpler definition. The main definition is already in your original statement of size dependence. Maybe it can be sharpened a little more, but almost everything is there.