The router will have a collet (1/2" as you've found out) and will probably not come with a shank (arbor). The arbor usually comes with the slot cutter. The slot cutter will _either_ fit on a 1/4 arbor, or a 1/2 arbor, but not both.
The arbor is a 1/4 or 1/2 diameter shaft that is threaded on the end. you install the slot cutter (and shims and bearing) on the end of the arbor and put the nut on it. Now you take this whole tool and put it in the collet on the router and tighten it down.
I think the slot cutters from t-moulding come with the arbor and bearing, etc.. so then you just need to make sure you have the right collet size for the arbor size.
Hope that helps.