What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras?

In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping algebra is given. Such an algebra is called an $h$-deformation. In chapter XVII the Drinfeld-Jimbo algebras are introduced and it is proved that they are $h$-deformations. Chapter XVIII gives some uniqueness results of these $h$-deformations (the rigidity theorems).

Along the way, $q$-deformations are also introduced (although there is not really a proper definition). One result states that you can view a $q$-deformation as a Hopf-subalgebra of an $h$-deformation. One major difference is the quasi-triangular structure. By definition $h$-deformations have a quasi-triangular structure, but $q$-deformations don't. By Kassel's definition, this implies that $q$-deformations are **not** quantum enveloping algebras.

An obvious question arises: Are there still rigidity results for $q$-deformations?

Further I'd like to point out that other good books such as 'Quantum groups and their represenations' of Klimyk and Schmüdgen introduce Drinfeld-Jimbo algebras using the parameter $q$. However, Drinfeld introduced $h$-deformations first. Whenever people talk about $q$-deformations, they seem to be interested only in the representation theory and seem to forget what a deformation should be.

Other than the above question, I would very much appreciate it if anyone could shed some light on these issues. In particular I am confused why some many people accept the $q$-deformations without rigidity results (or I can't find these). For sake of completeness I would like to state that I also asked this question today, which is related but focuses on something different. I am not trying to spam the site with a bunch of questions.

**EDIT:** I would like to mention that the Hall algebra approach to quantum groups seems to recover $q$-deformations, but it looks very unlikely to find $h$-deformations in this way (how on earth would you get $e^h$ from counting extensions?)