Halogen is a bulb type. Fresnel is a lens on a lamp. You could- for example, have a Fresnel HMI, but you can't have a Halogen HMI.
Fresnels, be they Arri or anyone else are the industry standard lighting equipment because the lens allows you to really control the beam and cut it with flags and get sharp shadows-- if you'd want.
The open faces, from Arri and others are good brute force lights--- normally used to give you a soft source or light up something in the background where you don't need the kind of control you'd get from a Fresnel. Also they are lighter (open faces) for the most part since they have no lens to worry about.
Another type of light is a PAR which stands for Parabolic Reflector. These are also generally brute force lights which have a long throw and are great to punch through some slight diff.
If you wanted to, by the by, use a red-head or a PAR on talent and still keep it directional, I'd look into using Opal.