Which router bit should I use to recess a door hinge using a template
This bit is called a "hinge mortising bit"* but it looks to be essentially like other top bearing router bit. 

The one difference between this and other top-bearing router bits is that the maximum depth of cut on this one is fairly shallow, which is what you'd want if you're routing into the edge of the door to recess the hinge. This particular one comes in several depths of cut, but to me, they all appear to be to deep for a standard door hinge.

The advantage of a guide bush is that you can use any old straight cutting bit and adjust the depth independently of the guide. With the linked bit (or a similar one) you're stuck with routing to the depth of the cutters, or you have to make a template the thickness of the cutter minus the depth of the mortise you're looking for.

* Amazon.com link - no specific brand/retailer recommendation implied