Sorry it's probably my fault for not explaining in detail.
Firstly Cat I is altimeter (it could either be a DA or DH depending on whether QNH or QFE is set). Cat II/III is radalt. I will try and clarify the rest by editing your post:
Precision Approaches (e.g. ILS):
DA/DH is used [yes always]
Minima can be underflown slightly [I know what you are saying so yes basically. In a geometrical sense you dip below the minima if you make the decision at the DA itself but in a legal sense you have not busted it because the requirement is to make the decision and intiate the missed, and of course we never bust minima]
Named Height (below 200feet; usually for Cat II and III) [yes]
Named Altitude (above 200 feet; usually Cat I) [minima may be down to 200' above TDZE either height or altitude, but higher is not uncommon]
Non Precision Approaches (e.g. RNAV, VOR, NDB):
MDA/MDH is used
Minima can not be underflown! [Basically correct. It used to be always be the case but now certain operators may use the MDA/H as a DA/H. However, you know if that applies to you because it is in the Ops Manual and you will have been trained for it. The general procedure now is to fly to 50' above the minima and use that as a DA/H (referred to as a Derived DA, DDA) and thereby still not dip below the MDA/H. Hope that makes sense]
Named Height (below 200feet) [No. There are no NPAs with minima below 250', certainly that used to be the case with VOR/DME and SRA (non-precision radar) sometimes getting you that low. I am not 100% sure that with all the new developments and re-purposing of terminology that something which is not technically a "precision approach" could have a lower minima than 250' but that is a grey area that should not distract from the big picture.
Named Altitude (above 200 feet) [see above]
The disitinction in all this is not whether you fly to a height or altitude. You can use either for both precision and non-precision approaches (the exception being Cat II/III is radalt height only).